Handbook of Hypergeometric Integrals.pdf

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HANDBOOK OF IIYPERGEOMET RIC INTEGRALS .- THEORY, .A,FPLICATIONS,

.

TABLES,COMPUTERPROGRAMS'

Series: Mathematics and its Applications

\

\L^.u,r* 4k^ * MATHEMATICS & ITS APPLICATIONS Series

Editor:

Professor G'

M'

HANDBOOKOF

BeU

Chelsea College, University of I-ondon in- their sc.ope' variety and Mntlrcrrrttics and its applications are now awe-inspiring its applications to the and pure mathematics growth in rapid rlrI)lll. Not oltly is there but new fields of statistics, and engineering Irnillliortrl fiulds of the phy;ic; sciences, and sociat organiiation' The user of ecologv in biologv, emerginj are ;;,;,ii;:;,i;;,, also leafn to handle the Sreat ,rn(lrrrrrrlics must assirhilate subite new techniques aird

and economicaliy' 1,,,*", ,rf tltc computer efficiently texts is thus greater than ever and our 'l ltc nced of clear, tot"it"

"J""horitative be comprehensive and ye.t flexitrle,. rrrii:x wilr endeavour to supply this need. It aims to areas and up-to-date mathematical new introduce will research recent Worhs survcying rvilt stimulate itudent interest by topics established on teits rrrr(ltods. Undergraduate series will atso include selected The day' present the at ,r,.i".tt"* applications relevant important topics to be presented' voluntcs of lecture no'tes which witl enabte certain ;;i;;,;;;';;;;,iJ o,t ".*i'" be possibre' to those who learn' teach' all these w?ys it is hop"j to render a valuable seruice In ,levclolt and use mathematics'

The Foundation Progranune includes

:'

Sciences iuathematical Mod;ls in Social Life and Management Institute of Technology' David N. Burghes and Alistair D' Wood, Cranfield Mechanics and Cqntrol to Classical Introduction odern M -'" prriJ Burghes, Cranfield lnstitute of Technology and Angela Douns' University of She f field. Vcctor & Tensor Methods Frank Chorlton, University of Aston' Birmingham' [,ecture Notes on Queueing SYstems tsiian Conolly, Chelsea Ccllege, London Universi!y' Mathematics for the Biosciences G. Eason, C. W- Coles and G' Gettinby, University of Stralhclyde' Tables' Computer Prograrns Handbook of Hypergeometric Integrals: Theory' Applications' Harotd Eiton, The Polytechnic, Preston' Muttiple Hypergeometric Functions Harold Exton, The Polytechnic, Preston' Computational Geometry for Dcsign and Manufacture Ivor D. Faux and Michael J' Pratt, Cranfield Institute of Technology' Applied Linear Algebra Ray J. Goult, Cranfield Institute of Technology' Generalised Functions: Theory, Applications Roy F. Hoskins, Cranfield Institute of Technology' Mechanics of Contifluous Media S. C. Hunter, University of Sheffield'

Using ComPuters University of London' Brian Meek and Simon Fairthorne, Queen Elizabeth College, Environmental AerodYnamics University of London' R. S. Scorer, Imperial College ofScience and Technology' and T€chnologists for Scientists Suroey A Physics of the Liquid State: of Wales and H. N. V. Temperley, Universiiy College of Swansea' University H. D. Trevena' University of Wales,'Aberystwyth'

HYPERGE.OMETRIG INTEGR.ALS THEORY, APPLICATIONS, TABLES, COMPUTER PROGRA

HAROLD EXTON, B.Sc., I\{.sc-, Ph'D' Tire PolYtecirnic, Preston

ELLIS HORWOOD LIMITED

fublishen

Chichester

Halsted Press: a division of JOHN WILEY & SONS Chichester 'New York' Brisbane 'Toronto

Cl^,*

'l'ltt ptrblisher's colophon is reproduced from

James Gillison's drawing

of

the

arr o and

(t.2.2.t) "-Y.,

rq

1.2.1 Convergence

tha

2l

E,CandHFunctions

Sec. 1.2.21

5.2.1 for

a more

detailed description of this function.

of functjon is the H-function of Fox (1961) This is defined in a manner similar to the G-function, that is A more gene;'al class

ay -Q+n

x

LLp,qt"l(b.Br),-.,(oq,uo),J = 2riiHm,"[_ll:,,1,],.,::r,:r]l LttL

(1

.2.2.I)

(1

.2.2.s)

(continued)

'22 (

lch.

Functions of One or More Variables

r

cont i nrrcd ) n

nt

.,( -)

n

I :-l

l'(b.-B.s') -l

l

l(I-a.+A.s) ' I l' j=1 II

p C: .. ;t f(1_6.+B.s)11 t-(a._A.s) j=nr+l' , , j=n*, 'J J

whcrc an cmpty product is interpreted as unity, O a m 1 p, 0 5 n < g, '))the A. and B- are all real and positive and the poles o{: the iltegrand of (1.2.2.5) are simple. The path of integration (l is a suitable contour of Barnes type which runs from -i* to i-, indented if necessary such that all the poles of f(b.-Uj.), j=1,..,m are to the right, and those of f(1-a *or=r, j=1,..,n to the left of the contour C. The integral (1.2.2.5) converges if TT. Iarg xl'1..; D, where

np

D

mq A.+ I B.- I B- .> O. j=1 .l j=n+t I j=t J 1=p+t J

= L A.-

Hypergeornetric Functions .of Two

]

Sec. I

.3

the

Lommel

Variables

23

.polynomial

2' 2,-ff'/2;n,-m,l-n-m;-x-) I^F-(l/2-n/ 3' Drdelyi et and the Struvb functiolt 2,.(1 :3/2+p,3/2t-x-/4) = /"/2 rF2 Erd6lyi et

(x/2)m (x), = j:* (,i,rn] Rfl,D'"

al. (195J) \'ol. II

page 35,

r:n

(2/x) ''P tt-.1x; , -p al. (1953) VoI. II page 38.

I.3 HYPERGEOMETRIC FLINCTIONS OF T\t'O VARIABLES

In addition to increasing the number of pararieters, hypergeometric. functions may be generalised along the lines of increasing the number of variables. Appe11 (i8BO) was the first author to treat hypergeometric functions of two variables on a systematic basis, and he defined'the four f,unctions which fo1]orv:

),

(1.3.1)

1.2.3 Special Cases

(a,m-ll(b,rn)(b',n) -m..n -G;EJGr,;tm'.nl " ) m, n=0 r

A short I ist of special cases of the general ised hypergeometric

function of one variable is

L-

now given

(q,.1 {a',nJ (b,m

'l'hc Bateman polynomials

(c,m+n) mln!

m,n=O

.Fr(-n,n+1 ,l/2+x/2;1,1;1) = Fr.,(x),

and

,Fr(-n,n+1;1,1;x) = Z,r(x) 2;22) = ,F^(-n;u+1 ,v+t+uf

Bateman (1933), Fo

+t##ilrf#@ ,-uJ'''(r), Bateman (1936),

the Rice polynomials

(a, b;c

,

c' ; x,

y)

r

(a

1'6 .

p+n)

The new functions are all.generalisations Appe11 derived them by consldering, first

uct of two

-^y

mn m+nl I ^Cff:fi;;fr-a'y' m,n=u.

.m..,,

" t '

(t ? )r rr'J'-' I I

(

\t

(1.3.4)

of the Gauss function. of all the simple prod-

Gauss functions

@

Rice (1939),

1,p;vJ = Hr",(k,p,v), aFr(-n,n+1,k; the Fasenmeyer polyncmials p*2Fq*2(-n,n+,,a',',ap; 7/2,1,b1,',bo;x)

;ri:i_:;llt,lrorr,

r(t

/ 2+n/ 2+m/ 2,L+n/ 2+m/

2

;l+n,1+m,

1+n+m ;

=f (n+1) I (m+t)

(z /

-x2)

*)n*^

Jr, (x)

J,r(x),

Erd6lyi et al. (1953) Vo1. II page 11, -tr

(i.3.s)

and replacing, in turn, each pair of products (4,m) (a' ,n), tor example, by the composite procuct (a,m+n), whbre both indices of summation m and n are involr"ed.

In addition to r:he functions F.5., FZ, F, and FO 8lven above, it would appear that we may also have the double series

the product of two Bessel functions ,F

,Fr(a,b;c;x)rFr(a',F'ic';y) = m, In=O^

iL ir,rn-n) (b,m+n) -m..n rc.m*nl mlnl^ / m,n=u--

(i.5.6)

Functioris of One or More Variables

24

[ch.

Convergence

r

A straigl-rtforward application of the binomial theorem, however, reduces (1.3.6) to the function 2F I(^,b;c;x+y); see Exton (I976)

=

page 24.

=

ano

=

I' ^_(a,m) m,n=u

(ar,n) (b,n-m) ib',m-n)nm..n (r.3.1.1) mlnl

mn rZ (a,2m+n) (b,n) xy r^ -n 7;'-;; k (d,n)m!nl _ _t_^(.,nr) lllr lt-u

(1.3.r.2)

of all these functjons is given in Erd6lyi et al. (19Sj) Vol. I pagc )24.

.i;A;J,[:] ;:;!' i*,*)

=

de F6riet (1926) Chapter 9.

li{a. -,

t-

, ]

m+nJ Ii (0. ,m) 'l j=1 l D

Ii (c .

j=L

)

,m+n)

1

j=1

1I

i

D

(bl,n)xmyn -l

;

(dj,*) l (dl ,n)mlnl

j I

The notation used on the left of (L.3.2.1) due Chaundy (1941), is mo:re compact than that used

de Fdriet

(1.3.2.1) to Burichanll end orj-ginally by

(r.3.2.s)

at greater length in Appel1 et

Kanip6

As described in AppelI et Kamp6 de F6riet (1926) page 396, Horn gave for the first time the general definition of hypergeometric functions of twc variables in which he stated that the double power ser i cs

E(x,v) where the

=*,1=6A*".,**rt,

(1-3.3.1)

coefficients satisfy the conditions

Am+l,n

It is possible to genelalise the AppeII frrnctions F, to FO so as to obtain the double hl.pergeometric function of higher order by increasing the. number of paianeters in a manner similar to the generalisation of the single hypergeometric function .in Section 1.2. This function was first defined and studied by Xamp6 de F€riet (1921) and is iramed after him; it has the following series representatron: D AB

fi,t

P _

or,n*,

(m, n)

Q[m'n]

, and ff=S*,,, ffi,D ,

RG;'

(r.3.3.2)

is of hypergeometric type; P,Q,R and S denote polynomials in the indices of -summation m and n of degree p,q,r and s.respectively. Apart from the compatibility reiation P(m,n+l)Q(m,n) _ Q(n+1,1)P(m,n) R(m,n+l)S(m,n) S(n+l,niR(rn,n) P,Q,R and. S may be chosen arbitrarily.

(1.3,3.5)

In order to investigate the region of convergence of (1.3.3.1),

Horn puts

Pfem.en)

0[m.n) = Lim e.* R(em,en)

anrl

Y(m.n) =

tls $f:ft:+, (r.s.s.4)

it is clear that O(m,n) is inflnite if p > r, identically zero if p < r, and is a rdtional function of m and n if p = 1. Let g = lxl ana n= lyl , and denote by D the rectangle in the positive quadrant of the plane O{n bounded by the coordinate axes and the straight lines parallel to the coordinate axes when

The AppelI functions

funct i on as fo]

Q.3.2-4)

1.J.3 Convergence

3.2 The Kampe de F6riet Function

Kampd

r[!l]:rr,

..,(a),b-Lr;., A*I'-ct (c) ;^)'

These functions are di-scussed

A list l

(t.3.2.3)

[l] ;,.,rt ,

and

0ther hypergeomctric functions of two r,eriablcs wcre investigated by Horn in a long series of papers ex,t-ending over the fifty year period 1BB9 to 1939. Here all the double h)'pergeonretric functions of the second .order and of two independent variables were systematically studied. In this workr as well as products of the type (a,m+n), we also encounter the types (a,m-m) , (a,2m+n) and (a,2nln) . Typical examples of the functions listed are

Ho(a,b;c,d;x,y)

c

-O:B;B',-:(b);(b'); (_ ';;;;;, ; i;j i ii, j i.,rr = uor([3]ix)s,rrp,

1.3.1 The Horn Fuuctions

Grla,a ' ,b,b';x,y)

RFc

25

I ows

are special cases of the

Karnp6

de F6riet

:

e = !/ Also certain Kamp6 de F6ri-et- func-tions are immediately reducible to generalised hypergeometric functions of one variabJ.e, such as

We

lo1t,oll

and

n = rl lv6o,t;

I

(r .3.3. s)

also take C to be the curve whose parametric equations are

Ilunctions of One or More Variables

26

6 = 1/o(m,n)

arid n = 1/Y(m,n)

[ch.

I

(1.3.3.6)

In so far as the convergencc of (1.3.3.1).is concerned, it now remains to consider the follornring five possibilities: (i) If p > r and q > s, the region of convergence consists only of the origin. (ii) If p < r and q I s, the region of convcrgence consists of the tvhole positive quadrant. (iii) If p < r and q = s, the region of convergence consists of, the strip between the axis Ox and the straight line n = t/l'Yfo,tl l. (iv) If P = r and q < s, the region of convergence consists of the strip between tlle axis Oy and thc straight line € = 1/ lo1r,o1 l. (v) If P = r atrd q = s, (1 3.3'1) converges ln that regionof the plane 06n common to D and C and which contains the origin 0. This last case is the most interesting in that , it deals with the complete (non-confluerit) double hypergeometric functions.

outlined above is now applied to the Kamp6 de Fdriet function. For this, with reference to the deflnition (1.3.2.1), suppose, foi: convenience, that B'=B and D'=D. It is evident that if A+B < C+D+l, then p < r and q < s, and so the series converges for all finite tralues of the variables x and y' If A+B. > C+D+l, then p > r and q > s and the region of convergence reduces to the origin only in the Oxy p1ane.

The general theory

The

or

greatest interest attaches to those functlons where

p=q=r=s,

A+B=C+D+1, and then

A_CmB .D_ I and Y(m,l'i = . A_C B-Do(m,n) = [m+n) lm+n] n

,

Three different possibilities now arise: (i) A=C, giving 0(m,n) = Y(m;n).=.1, and the region of convergence is lxl < 1 and lyl . t. (ii) A-C= -k. O, when Q(m,n) = *k7 1**n;k and v(m,n)=r,k/(*n)k, and the Cartesian equation of C is E-L/k+t)-l/k = 1.Thus C iies entirely outside of D, so that the series again converges for lxl ' I and IYI ' t (ii-i) FinailY, A-C = k > 0, when o(n,n) = (m+n)k *-k ,nd Y(m,n) = 1m+n)k n -k, ,rrd the region of convergence is now

l-

that l*'/ul * lyllkl . r.

1.3.4 Special Cases

Although hypergeometric functions of two variables occur in a number of applicatlons, the situations where they have been presented in terms of functions with indjvjdual notation occur comnaratively rarely. We now girre a few cases The Appeli polynomials,

a-c-c I F- (a,c,c t ;x,y) F^(c+c' -a-m-n,c+m,c r +n; c,c' ;x,y) = (t -x-y) flrfl'- ' 2' and

Fr(a.+r+n,-fl,-Dic,ct ix,y) = E*,n(a,c,c' ;x,Y), Appei

I et

Kamp6

de F€riet (1926) Chapt-er 6,

integral of the second kind,

the lncomplete elliptical

= Cosec0 F(OIk) Carlson (1961) , and the random flight prcbability function in two dimenslorrs, F

F

2, | / 2,1 LO /

o(

| -o /

-

4,1

/2;3/

2;

-3p/ a ;p/

r (p / 2) t (3 /

=ir2O,k2rin20

2,p/ 2-p

/

2

r

rlt

Dr

12,ult

)

12 :

Gp / a -1) 2.(t

'(arar) ' v''t(2-p/:) r"n-2_

-^l riarson (r944) page 4zr.

I.4 MULTIPLE HYPERGEOMETRIC FI.JNCTIONS

of generalising the Gauss function and its confluent forms by both increasi-ng the number of parameters and also increasing the number of variables may be car:ried on to any desired extent. To this end, Srivastava and Daoust (1969) have given the multiple series -t2(t,'t,arin)' o(n) Br n A 'J

The process

I

from which, as before; D i"s the unit square with one corner coinciding with the origin and with two sides lying along the positive coordinate axes.

such

21

Special Cases

Sec. L3.41

n r.tofn)*r-of')1xlt..xmn i ,1".* *.u!t)t i rlb(1)*r,or(i)r 'I nl r'l ' .j=t .-,-'-j .,-r-"i-t ' j=r' j

L

c

n

Dt

i ,r..* , *.q,!i)r i

j=*r''-ji]r"'i'j'j=,

.. rtd!1)**,0!l)f -1-j

n(n)

!..m.n ! n r[d(n)**-o.(')]m. nl't j=r'J

(1 .4. 1) Here and in what follows, it is taken that all indices of summation run from zero to infinity unless otherwise indicated. This extremely general nultiple hypergeornetric series is denoted by either of the two symbols which follow:

Functiorts of One or More Vari ables

28

,S

A:Br;.

'

C:D';..

A:Bl;..'B(n) ll1ul -c

C:Di;..;O(n)

I

[I The

r.l

r

,,r,' ,

.

1.4.t The lauricella Functions

,r(')i*

far the most important hypergeometric functions of several variables are the Lauricella functions:

l'^l

o

D'

I I rfa.l )' j-r _ t-r

C

D'

i=f

i=t

ffbl].. J

II r(c.) r r(dl). r

-l I

"1,

.otn.) 1:

tjn) rr,b1, . ,bnic1,. ,c" :xr,.,x,..,)

I

l

_, - t

J

I

(n)

{armr)

. . (arr,mr1)

the 0 ts,0ts,r.!rs and 6's arc l.ositive constants OT ZETO. It is clear that, if these positive constsnts are all taken to equal unity , then, for exampre, rl.+: u:1;';l:1 corresponds to the -{nl see (1.4.1.1) below. Lauri,cel la function F^'"', A

l) is certainly a useful generalisation, it seems that for .some pu rposes, multiple hypergeometric functions which are of a less gene ral nature are of more immediate value. To this end, we conslder the generalised Kamp6 de F6riet function first given and defined as by Karlsson f19731 ' Wh1le (1 .4.

.^,sku):(br);';(bn);* ' C:D l_(c) : (d,) ;. ; (dr,) ;"I'' "'i:

I

(br) ,mr) . . ( (b,-,) ,m,.,)xT1 . . xln (1 .4 .2) ((c),mt*.*mrr) ((dr),mr) . . ((dr,) ,mn)mr l .. ml n A more ge ne ral form of this function is somctimes emplo ycd +m,.,) (

)

:

I-A:Br;. C:D';.

-L

f

nr!..m

i)

(1.4.1.2)

I

,*,,. . ,,." ,,, : (b'); . l'][ (')[r.l: ',t::]] "J (d("'); (d');..; ;D

I

((b'),*t) .. ((u(")),mn)xft.'x mn n

(a,rnl+.+mr.,)

(c,m,

(br,rl)

**tt I m-1..m ln

. . (br.,,mr.,)*lr.

+ . +mr.r)

((c),m1*.*mr.,) ((d'),mr) . . ((a(n)),m,,)m1 1,,m,,!

.

(r .4 .1 .4)

four functions were first defined and studied by Lauricella [i893).If n, the number of variables, is made equal to two, these four functions reduce to the Appelt functions F2,Fj,FO and F, respectively: and jf n=1, all four functions become the Causs function ,F,. By means of approPriate Iimiting processes, a large number of possible confLuent forrns of the Lauricella functions arise. The most important of these are (b'mr) . . (b,.,,mn) xTr. .xln , (1 .4. 1 .5) oj") rur, .,br.,;c;xr, .,xn) =l (c,mr+.tmrr) ! m-!..m -tn m-m n x.l..x (b,nr+. +mr.r) ln (1 .4. 1 .6) vj") ruic1, .,cn;xl,.,xn) =I (cr,mr) . ..(crr,mn)m1 ! . .mn I

These

and o[") t.,u 1,.,bn_I,-;c;xl,.,*r,) (r.4,3)

(i.4.1.3)

I

rfn) tu,br, .,bn;c;xr, .,xn)

and

=i

rn-!..m IN

;B

( (a ),{..+mr.,)

(1.4.1,

I

(br,ryr) . . (brr,mr){1' .*ln

=I (.,,*.,1 . . . (c,.,,mr,)

where

(a) ,mr+.

-*ll

!..m,

l

A,B';.;B(') iir"l,0','.,e (n) I : i1u' ) : e' I ; . ; [ (b(n),' *(n) 1' *r' ' '*nlI '. 'r(n) C:D' [(c) :,1,,,.,U(')]' [(d') ::'] ;. ; [(d(n)),0.(n)r'

(

m,

(c,mr+.+mr_r)

ri Ird!nl) '

j=l

:j,(u,r::,.,]*T1.

(c.mr).. (cn,mn)

(nl (ar, -ri"' .,a,.,,b1, .,bn1c;xr,,,xrr.)

i r ru!') l

o

(u,"r...*,,)jurTr)

(n)

i-l t-\

and their Confluent Forms

By

I

,t(")1, [(d'):6r] ;.; l.(a("))

alternative notat ion is :-r )-!

29

[,auricella Functions

Sec. I .4.1 I

;B(n)l-*,.l

,r',. ,s(n)1, [(b'):$'] ;.;[(u(")) 'q("),. xr"'x

A

vE

lch.

See Exton (1976) Chapter 2.

m_

(a,ml+.+mr.,) (b,,mr) . . (br_r,mn_r)xil. .xnn (c,mr+

.

+mrr)

m,!..m _tn

m

!

[1.4.1.7)

Functions of One or More Variables

30

[(]lr.

I

1.4.31

Partial Differential

of a straightforward generalisation of Hornrs general theory of convergence as outli.ned in Section 1.3'3, the series representations of the Lauricella tunctions are found to be convergent with.n the following regi-ons: By means

l*rl*.*lxr.,l .

rl")

1,

,.B(n)

'l*1l,.,lxrl . t,

r[")

l/*rl*.*l/x,,1 . I [*rl,.,lx,.,l . r.

n [c,-(a+b.+r)*.]{.- o,,a=, irkk 'J ' J ' l'd*j

^o^,

1,.,1*n_rl< i,

fcr the rr,.,.'-to.

r[;),

(I.4.3.1)

(r.4.3.2)

(I-x )+ ' 'ax.I

'fhese four systems are particularly ifirportant in that thel enable these four functions to be defined for a1I possible values of the independent variables x1,..,xn, real or complex. The need for a means of carrying cut such a complete definition of the four functi-ons in question is the prime motivation of the investigation into the general integration of the four systems. This also has an important bearing upon the application of the Lauricella functions. In fact, the above remarks may be applied to all hlpergeometric fuactions of one or more varlables, but we confine ourselves to the Lauricella functions hcre.

of partial differential equations are:

.iil{h k=l

K

kt) (cont inued)

J

+

(1.4.3.1)

i "r r=i

x.

, lc i-(a+b+I).,1:= - (a+b*l ' -.'j '

have seen in Section 1.1.3 that the Gauss h1'pergeometric.function of one variable satisfies a certain differential equation. Lauricella functions of several varIn a similar way, the four:.l.4.1 arc particular solutjons oi ceriables defined in Section tain systems of partial differential equations. There are four of these systems, one associated witn each of.the four functions r(n), .(n), .[") and FSn) respectiyery.

I

o,

v rrt-xi) . .,3F2 *. iI.*la* Fi ' , *i( for the function -(rr) " a2ra, K=I K dX. klj

and

We

*:(,-.j)#

ab.F =

ktj

1.4.3 Systems of Partial Differential Equations

for the function ,1"),

-

+ .[c-(a.+b.+I]*.fS- - a.b.F = 0. J,a*j ) ) I :

case of o["), *" have convergence r+lrerr l*1 xn may assune any finite va1ue.

t,

3I

J

The series representations of the functions 6j") u,',6 n1") converge for a1l finite values of their var:iables, and in the

These systems

Systems

(cont inued)

1.4.2 Convergence

and .S")

Sec.

anJ for the functio, aS'),

I ^r )- 1.:1) x,lKdxi, N_

1

kti

n r

^2dF

"s3x s=L r 3x s sl-i when r=j YIY

abF

L.

-0

(1.4.3.3)

N

n d^2-r 2r- * \ f -. x.[1-x'1d I ) 6t-*. -rL Koxr.oX. 1 1^z I r ^,:-----^ dx. )

n^_ its b.)x.i:-ab.F=0. kax) K . k=l

N-a

^

klj (t

.4

I

.3.4)

ktj

In the above four systems, j = 1,.,n. It has been shown by Lauricella (1893), that the general integrals of the systems satisfied by the functions FIn), F(") ana r[n) depend linearly upon 2n arbitrary constants, whi-1e that satisfied by p(n) depends Iinearly upon only (n+1) arbitrary constants.For discilssions of these part ial differential systems in more detail, the reader is referred to Lauricella (1893), Appe1l et Kamp€ de F6riet (1926) page 117,Erd61yi (195O) and Exton (1976) Chapter 5. rest of this book is devoted to a study of various types of integrals of hypergeometric functions. These integrals are of importance because of the fact that a.1arge number of special functions of applied mathematics are of hypergeometric form, and integrals which involve special functions are of frequent occurrence in such fields as mathenatical physics and statistics. A few examples are given in Chapter 7.

The

Gauss Function and its Confluent Forms

Sec.2.1.ll

33

1

Chapter 2

I Hence, /*u -' (, -*) b- 2F1 ("id: r*k) d*

Integrals of Euler Typ"

. [t'd'afk'(a+1)/k''''(a+k-1)/k = I'(a)r(b) iG;E - z*k'i*kL f , (a+b)/k, (a+b+1)/1 ,.. , (a+b *t-r)/k;sl where

2.I GENERAL EULER INTEGRAIS

Euler integrals constitute an jmportant class of finite i-ntegrals and the general integral of this type may be rvritten in the form 1 c-] * - (1-u) h-" f (.r) dx. (2.1 . 1) I

lx 0

If the function f(x) in the integrand i s capable of expansion in a power series such as f

(xl = I'^ . n- ' n-{i "'

(2.r.2)

then, provided that .the radius of conver-gence of (2.1.2) is not greater than unity, we have 161 -1 , = F c^ jx ? arn-I,, .b-l dx. (2.1.3) ^,, dx jx a.I-,rl-x).b-I l(xJ rl-x) l. n0"0 o The inner integral on the right of (2.1.3) ma1, be evaluated as a beta function [Erd€1yi et a1. (1953) Vo1. I page sl B(a+n,b),

(.2 .

t .4)

provided that Re(a) and Re(b) are both positive. The Beta

function (2;l.A) may be written I(a+n)r(b) -_ r(a)l(b) I

as

(a,n)

fG;D-'G;6,,I'

rG;b+,)

so that

;l

0

.-

/a r r\ (-'1'J'

* (a

nlc ='l?]i*i,,lofffi.

e.1.6) This general result may easily be extended to cover nultiple series representations of f(x), and has very many spbcial cases involving special functions,

i*'-',,-*1b-rr1*1

2.1.1 The Gauss Function and

ib ConIluent Foms

If f(x) is of the form of a

r(x) = rF,(c,d;r;s*kl

Gauss

k is a positive integer and Re{a) and

=,i.ft*#P(=*k)n.

(2.1.r.i)

r)

1

| .2)

This result may be expressed in closed form when the hypergeometric function on the right is summable. If r^,e put c=f, the Gauss function takes the form of a binomial function giving the result 1

/*'-' (t-*)b-1 1t-sxk;-d d*

=

0

Re(a), Re(b) > O. {2.r.r.3) If,furthbr, rr'e put k=1,.we obtain the xe11-known Euler in tegra I formula for th'e Gauss function, .see (1 . I . 3 . 5) , 1r..r-f f(a)f(b) c,d,3t.l , = -ff;+bJ'-2Fi[s15.sJ, -,b-1,.(i-sx.).-d dx {2.t.1.4) lx ,(a-x) oR" 6r; , Re (b) > o. A numbei of -special cases of the Z*kFr_k function on the right of (2.1.L.2) mav be expressed in closed form using the varlous surmation theorems for the generalised single hypergeometric f,unction. See Slater (1966) Appendix III. We now give 3 few examples:

1.,,

,-r

(2.I.r.s) l*^-' (I-*)o-'"F, '0 zt r.,d;r;sx)dx = +fl+P The Clausen function of the previous expression may be reduced in a number of cases, some of which are (i) a=f, b=d-f, (ii) a=f, s=1 and (iii) a=-n,b=d-f+l-n, s=1. In case (i), the right-hand member of (2.1.1,5) reduces to a binomial function, and we have l, f -r .- -d-f-l rf)rrd-f) (1-s) . -f- (2.1.1.6) 'tr-x)"' 'rFr(c,d;f;sx) dx = ri6i---r l*' o R"1f;,Re(d-f) > o, . r.

.rrri'j;fi:'l

lsl

In case (ii), the integral concerned may be evaluated by applying Gaussts summation theorem fSlater (1966) Appendix III], and we

n""

= Ilfi*l:i+?#tr-#, ;*r-,1r_x)b-1rn,r.;di,rd,. .1 r; l[r+DJl(r-cjttr-uj 0

function rF' then

Re( b)'0.

e

i.r.7)

Re(f),Re(b), Re(f-c-d) > 0. Saalschtjtzts theorem is now applied to the hypergeometric function on the right of (2.I.1.5), when case (iii) yieldS the result .1 c-1 A-€rl-n " "2F,(c,d;f;x) dx = J; -"'(t-x')" (cont.) (2.1.r.8)

lch.

Integrals of Euler TYPe

34

2

(cont inued )

=,I

(al I' (d-f+ I -n) I (f -d+n) I (f-a+n) r (f) r (f -d-a)

Re(a), Re(d-f+1-n) , By applying

same

the formulae

we have

1

the

.

(2.1

.1

.8)

O.

a+k-l

.l

f^a ;,1 ,"..= +AHP /*'-'r,-*)b-1rF1(?;,*k;d* r*^L.,uib,.,*T*, r*r.Fr*rlt'I::'-T, r(a+b) ,** f;".. o 1 (2.1.1.s) 1-

una j*"-'

o

1t-*1b-10r1 (-; f;

r*k;d*

t

a

I

ix 0

2. 1.2

+t

a+k-l . -l 'kl a*b*k-1. sl

=H#Pr.'r.,l.li; r' k ' ''-- k-' l before, Re(a) and Re(b) , O. If we Put () I. I.9) reduces to ,r-€_r ,_ _= r(f)I(f-a) e, ^tFt(d;f;sx)ox ^s Ir -x)"' ffi Re(f),Re(d-f') > 0.

where ,3S

b=d-f

Firstly, we have r [-. ,*b a+b+k-l. i a-L..b-t l'' k 'r' k ' ,,.!o- _ r(a)I(b) (1-s) (t-x) Jx t*fFll a a+k_t I (a+b) o

LI'''

k= l" ,

,

I+ a b,

' t i*ld* 1*a-c;

I

_J

'

(2.1.1.r0) a=f and (2.1.1.11)

I (1+a) r (1+a-b-c) f (i+a-b-d) f (1+a-c-d) provided that Re(d),Re(a-2d) > -1 and Re(b+c+d-a) > -1. Tf either b or c 1s a negative j-nteger, the third condition of convergence of (2.1.2.5) may be relaxed because the hypergeometric functions involved arc terminating. 2.

1.3 Double Hypergeometric Function

l{e now consider integrals of the form l-r-.l,, ..,b-1 .C:D;D' r(c) : ( O, in o.rder to ensure convergence. to

If the Kamp6 de F6riet function is expanded as a double series, essed in the form ( (c),m*n) ( (d),m) (4' l,!)Imin 1 ! L

Re(b+c+d-a-1) ,

0.

(t+a-c-d)' (2.r.2.3)

(2.r.3.1) (2

2)

(2.t.I.3)

inside the integrand of (2.1.3'I) then this integral ma1'be expr-

ri*r*k (**n) -, ( r-*)b-Id*. (2. r.3.4) J* For the process of interchanging .ihc operations of double sunnation and integrati-on in the previous expression to be justified' the Kamp6 de F6riet function concerned *rrrt .orrr"rge for l*lSt' The inner integral of (2.L.3.4) may be written in the form of a beta function, so that (2 . i . 3. 1) becomes (2 .1 .3 .5)

aFO

-(1+a) r (1+a-b-c)l(1+a-b-d) f

| .2.4)

O and

2

A straightforward generalisatiorr of (2.1.L.4) may be written

l'{any sumrnation theorenrs may be employed

;

_ r(d)f (1+a-2d)I(1+a-Ultft.a-cltCt-a'

Generalised Hypergeometric Function

i*'-'t,

k

ls| . :.. Secondly, if the parameters of (2.1.2.2) are suitably specialised, (2.1.2.3) can be applied, and we have

where Re(a), Re(b)

nethod as was employed in deducing (2'1'L'2)

Double I{ypergeomctric Function

Sec. 2. I .3l

where Re(a), Re(b) > 0, k is a positive integer and concerned are elther convergent or terminating.

at1 the serles

[Ct.2

lntegrals of Euler TYPe

36

If we treat the integrals (2,1 .3. 2) and (2.L.3.3) similarly,they may be written, respectively, as a+k-l. a C:D;D '+k[-(c) : (d) ; (d') , kr"r.r-T-, r(a)r(b), (2.r.3.6) -T1"-6I-' a+b+k-1. ','-] a+b --7-, (s') . (f) (s) : F:G;G t--1:-,

I

I

'+kl

;

,

KA

a+b+k-

(

1

I

I,SI .

o

I

Anumberof simpler forms of (2.I.3.1) to (2.1.3.3) Suppose that we tet C=F=k, cr=(a+b)/k,.,cO=(a+b+k-1)/k and fr--a/k,.,fO=(a+k-1)/k in the first of these expressions. The becomes

1

.3.8)

splits up i.nto the product of a pair of hypergeometric functions of one variable givir.g the result

whj-ch

rl:B;8,[f..

,'.li

u.

1;k+1 )*u-r 1r-*1b-t ,F; 0;k 0

[r"t I ., a+klr

[,,,

r(a )r(b

r( a*b) c*tFG((c),d+d'

(2.r.3.e) obtain the formula

; (g) ;s)

0

=

I(a)f(b)

ffi

o

a

L,r, 'k'

o*

'

cFc*k((c);(rl' k

'''

k

:

I:l:l]

;',,.u

(2.1.3.11)

o*, (2.t.4.1)

r,-.tlo.

r1-*rk,,snX

(2

(c): (d' ) ;' ; (df:]]'=1, .sn-,,,n*t ]*u-t (r-*)b-1 .c:Dl 'o'Gf{r),

{r' l;.;(g'

al tu

[')

r (a) r (u) o*x 'r:G+kl_1r1 I (a+b) "c,

);

.1 .4 .2)

dx.

I

(d'),h,. ,h;.; (d("-1)),h,.,h; , (n-1) ' : (g'),h,. ,n;.; [g' ),h,.,h; -l :

-1',

,=:l

(2.t.4.s)

In the expression (2.1.4.5), the dummy parameters have been introduced in order to preserve thc compact notation of the generalised Kamp6 de F6riet function. integral (2.1.4.2) is slightly different from (2.1.4.1) and (2.t.4.2), and will be discussed in a li-ttle more detail' As in the previous section, it is assumed that the multiple series are

The

a+h a+b+k-L;r+s).

l

J (2.1 .4.3) ious sections of this chapter, k is a positive As in the p integer and th e real parts of a and b are both taken to be positive. The in teg rals (2. 1 .4. 1) and (2. 1 .4.3) are straightforward general isat icn s of (2. 1 . 3.5) and (2. 1 . 3 .6) respectively, anci so we may stat ei rnmediately that (2.1.4.1) and (?.1.4.3) are equal to, respect ive 7Y--- a -l a+b-l r(a)r(b) . C+k ,nl(.J,[,.'.., U-' :(d');-;(a(n))' ' - s' r(a.E F+k :Cl -^. a*b a-b+k-' (e');',(e{n))'"''''t K I " (2.r.4.4)

(2.1.3.10)

*,,r,_,.lt

of(c)

-,,,,*u-l

'

0

,, =.! u.

Compare Exton ().976 ) pag e 15 The expression (2.1 .3./) giv

i*'-'rt-*)b-ttF,?i

' !'::]]'.,*u, (g')] .;(c''); 'L(r) lt

'-|(r),;,'

= rfiHP r'.,[i]1,)r,,.' r[!l]:'r D=1,G=O,D =k+1,Dt=k and

F:;:[!:]'

i, D[c) : (d') jx r-i- (I-xl-b-] "C: ' t't 0 [{r) , {s, ) una

(2.

In (2,1.3.6), put r=s,

Results of the type (2. 1 .3.9) to (2.1 .3. i1) may often be expressed in closed form if the parameters and variables are suitably specialised; the various summation theorems of the generalised hypergeometric functions of one variable are employed' See Slztter (i966) Appendix ll I.

)*u-' 1,-*;b-1

(2.1 .3 .7 ) may be deduced.

i.'-',r-*)b

I

in this.section, Euler integr a1s of the generalised Kamp6 de F6rlet function IKarlsson (19 73)l are considered. The following integrals wilL be evaluated:

(e)

(g')

function (2.1.3.5)

-1

2.1.4 Multiple Hypergeometric Function

(c)

r(al I (b) - f,;p+k;Ir+k [ -...^.-:r I (a+b) F+k: G ; Gr Ltt''u*'

Multiple Hypergeome tric Func tion

Sec. 2.1 .41

lntegrals of Euler TYPe

3rl

lch.

2

either convergent over the appropriate ranges of their variables or tlrat they are terminating, The multipLe hypergeometric function of thc intcgrand is expanded jn. series of its arguments and (2.1 .4. 2) rnay then be written tmr,) ((d'),mr).. ((a(n)),mn)sft..smn ,,((c),m,+. . L,

"((f )

, nr

,

* . *rnn ) ( ( g ' ) , nr ) . . ( ( g

(")

)

, mr.,) m1

I . . mn

I

(2.1 .4.6)

o*.

0

beta integral involved in the expres s ion cons iderat ion. This integral may be written in the forn

0rrcc again, we have rrrrtlo::

l'(0)

I'(b)

(a,kmr+.

l'(rr+hl)

n)

(b, kmt * . +knr.,)

e+h

-a+b+l (f,tn,*.*km,r)- (f,kmr*.+kmn)

4-krl-.

-kmn

,

(2.1.4.7)

provided that Re(a) and Re(b) are both positive. The required rrv rr I uat ion then fo I l ows : -1 a+k-l ..rr,. r-.. a ,,(nl.')i.. (c), :(d');.;(d , r,(a)r,tbl_c+2k:ol f, k = - (ir-G)--+F+2k:Gl /-r r \u v/ " ' . r.,. '1,(f), ,...., a*b,. ,a+h+k-l ..o J i. ; (gt"'l;ot"'otJ k k I

(2.1.4.8) Cases

A Iarge number of EuIer integrals involving special functions may lrc deduced from the expressio;rs given in Sections 2.1,1 to 2.1.4. (lcrtain representative examples are now obtained. Ileplace the Gauss function of the integrand of (2.1.!.5) by the corresponding Jacobi potynomlal 8nd the following result is obtained:

i*'-t 6-n

Re(a), Re(b) > O. . i.6) , we obtain polynomiai I n+l ^ Pn(I J (1-x)" From (2.1

a

|${fif;ffiP

.s.2)

2 x v) d x

0

Formula (2. I .1.1O) may be e x pr e S SC t he Bessel function, when it t a k I

. o-l h-l ' J (sx) dx Ix" -(1-x)" .C

n

"

Re(a+c),Re(b) ,

O.

=

2

tr

I a +c

I ( c+ 1 , r( (

6 -c'-'

(2 .1 .s .4)

ffiJs1c*t1

k=1, I

t a-I (r x -- -x.). b-I Fr(c,d,d';f ;rx,sx) dx

I ot

l,l,l.l . t,

j

r(a)r(b).2:I;Ir^., l,d;d' !r.s), Re(a),Re(b)ro. (2.t.s,s) - -f(j= 2:O;Otf.a*b:-;- ;" If we let a=f and b=c-f, the above result reduces to I .-, n-r-l r, (c,d,d' ;r;rx,=*l ={*#P(t-r)-d(t-r)-d', {*t-t,I-*)t-t-' " (2.I.5.b) Re(f), Re(c-f) > O. in (2.1.3.5), so that, if k=I, we have r, -.b-1- F5,c;c"-,(g)l(g');^"'-\) rO:D;D' r-: (d) :(d');-. . dx Re(a), Re(b) > o. 1*-'fI-x)" 0 r(a)r(h)r-l:D;D', a : (d); (d"' (2'l's'7) -= lft+5J_ 1:G;c'(a-b:(g);(c'1;t't)' that

C=F=O

The Karnpe de Fdriet function on the left of (2-1.5.7) may be wri-tten as the product of a pair of hypergeometric functions of one variable. Many snecial forms may be deduced from this, for exampie, we may write -l t, a-r (b) -1 : z;zl a . b-1 z'2'I'.i;r,'1, K(rx) K(sx) aIx- 'n-x)-

--

' ...

=i-f-fr -,rs.v/

b

' -'Y -) - ,-n.n+c+d+l.a: c+l ,a+6 31 2l (2.1.s.1) re 5 u 1t involving a Legendre

tr-*)b-I R''d61-2*y;u. =

If b=l and a=c+2, we have the simpler result 1r.."-r lx Jr r-..\ tsx) dx =- z(t*1)/2r([s"*sl/zl /zG)' provided that Re(c) , 0.

Now, suppose Lt-u-l

I

2.I.5 Special

39

SPeciaI Cases

0

a

+km

I

A few examples of special cases of integrals of Kamp6 de F6riet functions are now considered. From {2.1.3.1), we have, on putting

1

, ]*r*k*l*.*k*.,, g1-*1b*k*I+.tkmn-I

Sec. 2. I .5

a

J,;l .-r,--l

Re(a), Re(b) > O, and K(rn) is the complete elliptical

1 'l,l'] i,1,iI ,.|"-rb; t ; t;

I (2.1.5.8)

integral.

This section is concluded by mentioning a few special cases of Eule: integrals of the generalised Kamp6 de F6riet function (2.1.4. 1) tg Q.l .4.3) . The Lauricella function rj") i= of frequent occurrence in a number of applicatlons, see Extcn (1976) Chapters 7 and 8, so that rve give three different types of intfor*ulae which involr'e this function. If we put C=D=F=k=1 "gril and G=O in (2.1.4.4), we have t ][*u- 1t -*1b 0

-' j"l (c o

,

d, , . , d,", ; f;srx,.,snx) dx

= (2.1.5.9)

(cont i nued)

lClr. 2

Integrals of Euler TYpe

40

(continued)

(2.r.s.e) = HHPrf:Iri;,iuid1;';dn;s,,.,sn), Itc(a), Re(b) > o, lrrl,:,lrnl . r. 'l'wo special cases of this r:esult now fo11ow.Firstly, suppose that f=c, when the FSn) function on the left of (2.1.5.9) -splits up into the product of n binomial factors and we obtain the Picard integral for the Lauricella functj-on njn) itsetf. See Lauricella (rBe3).

I (t-sn xJ-d, 7*"-'(r-*)b-l (1-s,x.1-d...

61

=

dx

' ,o n;a+b;s1,,,sr",), (2.1.5. 10)

of (2.1.5.9) also reduces to ftrnction if a=f and b=c-f, and so ) r-r - . c-f-I F;"'(.,dr ,.,dn;f;stx,. ,s n-x) dx i*^ -(l -x)" - -(nl

The right-hand member

ur', njn)

r .-.^

'dl'''d,ic;st'

"s,,)

(2.].s.11)

'

(2. 1. s. 12)

case

I

.J.*"-l (t-*1b-Ia(tt)

(c,dr,.,d.,_],a+b;f;rl, .,rr_r,sr.,x)

dx

0

=

(2.r.s.13)

+e#?+i") {''or,''d,,*r'a;r;sr"'s,,),

l{c(a), Re(b) > 0 and ls1l,.,l.rl 'l rr rrr i ng to (2.I .4 . 8) , we have

. r.

(2'1's'rs)

integrals of Erler type which are of lnterest involve the function 1F1 or the Bessel function, which may be written as a gF1 series. We begin by considering two general integr.als cf the type I I '-' (, -*) o-' t, (c ; f; sxk) f (x) dx (2.2.1) rx , Kummer

'

-

.-,.,, c ,+

k.. a-i ., -b-l [* "'(1-x) oFr(-;l;sx'')f(x)dx,

I

where Re(a), Re(b)

>

0 and k is a positlve integer:.

It, is now assumed that f(x) can be expanded as a power series of the type (2.).,2) whose radius of convergence is not less than tr. The integraT (2.2.1) may thus be r'"ritten ol

(2.2.3) ;.I,n/*u*n' (, -*)L-irn, [.; f;r*k;dr:, n=o "o uhere the process of interchanging the summation and int egrat ion is val id on account o.f the assumed convergence of th e series expansion of f(x), The integral of (2.2.3; nay bc eval uated by means of (2. 1 . 1 .9) . The integral (2.2.1) may then be written as -l i: a+n a+n+k- I * (a'n)hn l'' k r(a)r(b) ' (2.2.4) t*ktsz+kl ^ a+h+n a+b*n*k-1 sl' TTilE)*,,l6G; I

L'' * "'

;-l

k

the simpler result n)(c.nlh r-r"/"m-sn I(a) I(b) .(a'm+"u -r(a+b) r[3i6,m+n) (f,n)nl' Similarly, using (2.1. f.10) , the integral (2.2.2) ed in the form ,"{L', '-l r(a)r(b) - (a,n)h,-, - [ 'fl, a+b+n sr[*n+k-1.51 tTa-Ff-l^1a*E,nI k''k-,zl ' " n=u we have

'

dx

t

(2.2.2)

0

If k=1,

/*"' ' (t -*)b-' ol"' (c,d1, . ,d,,,;f ;srx[1-x], ., rr* [1-x]) 0r(a)I!b)+s:t[-c, a, b :d,;.;dn;s, rJ a+b a+b+I^ Lr' ;-,-1 ' LL

'

2.2 EULER INTEGRAIS ASSOCIATED WITH THE CONFLUENT HYPERGEOMETRIC F I,'N TION

nt

I

r;le*u)-s:ol.

' 'sn)

j

nl"l (.,d1,.,dr;f;sr,.,sn_l,s,,x) dx /*t'-11r-*;b-' i)u - r(a)l'(b)rl,rit'di'h;';dn-1'h'dn''irIt.b)---+'t:t[f: h ;.; l ;a+b;=r'''td.l

its special

,

and

Sinrilarly, (2.1.4.5) gives the expression

and

-- r(a)f(b) rG;El- ,'1;ita+ot i..'l'o'.'.'i'o';sr' rI ;.i rn t Re(a),Re(b) >.0 and lrrl,.,lrrrl . t.

dx

0

0

r(a+bi

]*^-r 1r-*;o-tro, (.r,dl ifr;srx) . .2Fr(.n,dnifr;sr.,x)

lt{any

+A#?+j') {',0,,

l_(a)l(b)-(n').., r; '' (f

4l

Integrals Associated with Confluent Function

Src.2.2l

(2.1.s.14)

_l

l(r'(rr), l{r.(lrJ > O and lrtl,.,lrnl . a. llnirlly, llrr. l'rrnula (2.1,4.4) is specialised so that an integral nl I lrr' yrr.oilrrcI o{' seve::aI Gauss functions is obtained: -

Lt,--k-,.,

k

(2.2.s) may

be evaluat(2 .2 .6)

;_l

(2.2.5) and (2.2.6) are important because they include many integrals involving special functions.

The expressions

42 2.2.

lch.

ntergrals of Euler TYPe

I

2

I

Ceneralised Hypergeometric FuncCon

Double Hypergeometric Function

$ec.2.2.21

2.2.2 Dovble Hypergeometric Function

I

If f(x) of the previous section is of the form of a generalised hyJrcrgeometric function of one variable, we have the two results wlr i clr fo 1l ow :

Consider the intcgral

).'^rL o-'1Fl (. f (d) ; (g) ;rx u) o* ; ;sxo; ./*"-' ( r-*) DFG( o Ta a+k-i :c;(d)

0

t(.r)r(b).k:I;Dl :,.,+ K ^ fft+b.)-k:l:Gl " ' ^ '-1 a+b a+b+k-1 .

==

LP

luttl

rr

1,,,-t-- -b-1 k /*"' (t-x)"'oF,(-;f;sx")rF.((d) 0 r'(a)r(b).k:o;n[ 3, .,+l

; (g)

:-;(d)

l1a-t I_k:1;Glroo ,*l*r_t ^ Lr.,., u :ti(8J

wlrcle

Re

(2.2.1 .7)

1 k-. rx

Jdx (2 .2 .1 -2)

S,

(a) and Re (b) > 0.

Sirrrpicr forms of the two previous results occuf iF r takes c:crrtai.n particular values. For example, if we put k and r both crclruil to rinity'1n (2.2,7.1), we obtai,n the result

l, r-f -' (--I -*)-b-I "-' rF, .[*u

{c ; f ; sx) DFG ( (d) ; (e)

(.rTl=i - . F^ . [a+m, (d);a+b+m, (e];tl (f ,m)m! D+l'C+IrmlOG-U,m) i

(",t,)

,

(2.2.r.3) artd further specialisaLion of the OFa function may result in a form of the inner D*1FG*1 serles of (2,2.1.3) which can te expressed in closed form. Let D=2,G=1,dr=-* and 8=dr-N-brl, where N is a positive integer. Saatschtltz's theorem may now be used to sum the inner series in the form (a+b-d.N) (a+b-dr+N,m) (a+b,m) (a+b,N) (a+b-d.m) (a+b+N,m)

(2.2.r.4)

llence, I 0

(, -*)

D- I

(c

rF, ; f ; sx) rF, I (a) r (b) (a+b-d.N)

/*'-'

(d

r,

-N ;d

r-b-N+

i*,-r 1r_*;b-i,r,

{. ; f ; sx) r3

;l;i I, [i] ; [;] ; [ ] ;,., tx)dx. :

I ; x)dx

=iG;51-Ie;6;Ni3Fa(a'c'a+b-dl*N;f'a+b-d.a+b+N;s)'(2'2't'5)

I(a)l(b).(3), -r(-a-)trrte also have l,

a : :-:(d):-:c;(e);(e');c \a*b::-: (C):-:f ; (h); (h');r'r r ' rl"

1)

(2.2 .2 .3)

.b-] ^ (.i1;sxjl-G:H;H't(8i ^ --D:E:E',(dJ (e) ; (e' 'rx,t)dx ^-l-[l-xJ (hJ ; (h' IF] 1* (d):-:c;(e);(e' ;s,r,t1 r(a)r(b).(3).,' (g):-:f;(h);(ht | (a-,IE and

(ei ; (e' (h) ; (h'

-::a:(d):-:c;(e);(e');< () _ r(a)r(b).(3)r *a+b::-:(g):-:f;(h);(h'j;s'r't') r rr ' l2'2'2'5) f(a+b) Other results of this type may be deduced' Special cases where, for example, Appe11 functicns are invoL"'ed will be disctr.ssed in Sect'i

on 2 .2 .4 .

2.2.3 Multiple llypergeometric Function

vast majority ofthe special functions ofmathernatical physics *ry be expiessed in hypergeometric form. The most convenient generalisation of the hypergeometric function of several variables js the generalised Kamp6 de F6riet function of Karlsson (1973). We recall that this function possesses the fo11owing multiple series repreientation:(n),. -A:Br(a):(b');.;(b' 'l;, vl '5'D'1c) '''"n/ : (d');.; (d(n)) "'I ( (a),rnr+.*mr) ( (u'),mr) . . ({u.(n)),rn,.,)x?r. .xln (2.2.i.).)

The

and chemistiy

_T

((c),mt+.+m,.,) ((d'),Tn1).. ((d(n)),*,,,)*1 I ..n

!}

(2.2.?.

result similar to (2.2.5) where f(x) i-s expanded as a-double series holds, so that (2.2.2.1) takes the form l(a) l(b) 1(a,m*n*p) ((d),n*p) (g,In) ((e),n) ((e'l,p)smrntP ,n)((h'),P)mln!P!' fTa;51- r (2 -2.2.2) This result may be expressed in terms of Srivastavars triple hypergeometric function ISrivastavb (1967b)] as follows:-

t, u-r-(1-x).b-I 1F1 (.; r, =.t o!:fi :fi i r[!] Jx 0

;.x) dx

0

r,(P) = 1.1?) -i(a+b)

t-

A

S,

r. r.r 'r'\6r

U

+3

l_

I

Integrals of Euler Type

44

[Ch.

2

If A=C=O, this function reduces to the product of n generaliscd hypergeometric functions of one variable BFD. On the other hand, if B=D=O, we have a single.function AFg whose argument consists of the sum of the arguments of the generalised Kamp6 de F6riet function frorl which it is obtairred. Furthermore, if B=1 and D=O, a reducible forn of (2,2.3.1) occurs when all its a:^g:.lnents are made equal to each other. It then takes the form . -F-t(a),b'*.*b(n) ; (c) ;xl A+l'C " general Three types of Euler integral involving the generalised Kampd de Fdriet function and a confluent hypergeometric function a.e considered. These are

i.'-',t

-*)b-I, r, {. ; r;sx) t3;i,i:] E,.

(d)

(e : :

[

:

]

(e')

'a,n.,*r,,r.,,, "D: .D: E[(dj

'.,nItrt : (h') r,.,[1-x],., h?

t

(n)

trr*, . ,r,.,x)dx, ); e.2.3.2)

(n) ] (h

, (n) [e' tn

(n),

'.,

, l*,r

2, . , rn) dx

(2.2.3.3) tn)-(e'-J 'rlx,.,r.kxJ Ih

(n)

)

.,.,Ir-*f

;

a*.

{2 .2 .3 .4)

integrals can be evaluated by us ing the multi-dimensional extension ot (2.2.6). Hence, (2.2.3.2) may bc written as (a,m+mr+. +mr.,) ( (d) ,m,+. +mn) (c,mJ ( (e'Lmr) . . ( (e @ ,mr.,)

t inued

( con

,

)

(

(e')

,mr)

( (e

(

[h')

,m1)

(

(n)

;

m-m

,

n,.,) sm rll..rnn

smm, rll,.rnnm " mln\l..mj The

( (h

(n)

)

,

results (2.2.5.5) to (2.2.3.7) require that Re(a) and Re(b) are both positive .for convergence of the integrals concerned. It is also understood that all the multiple serres concerned ate either convergent cver the range of integration or that they are terminating. some of the special cases of the various Euler integrals inv6tving the confluent hypergeometric function and othel types of hlAergeometric functions wj'11 be discussed in the next

The

sect i on. 2.2.4 Special Cases

A large number of Euler integrals involving special functions may be obiained by appropriate specialisation of previous formulae i-n Section 2.2 and it-s subsections. The generalised Laguerre polynomial and the Bessel function of the first kind may be expressed as hypergeomet::ic functions by means of the the foiiowing formulae respectivety Isee Erd6lyl et a1. (1953) \'o1. II pages l89 and 4l:(a+t,n)-F- ( n;6+1;x) (2.2.4.1) rarxl -n\^' =

nl

{2 .2

'

(c,m) (a,m) (b,mr)

.3.s)

((e') ,*1)..11e(n)),*r)

(f ,m) ((hr),mt).. ((h(n)),mn)

m rr l ' 'rnn smm. ' mlmr !-mj

12.2.3.6)

and

-rG;61-r

2

;-Uo t

If these expressions are substituted into (2.2.1.1), I

/x "-

I

11

-*)

b-1Ld

{sx1 -,.

[,/(rx)

fr.2.4.2)

we have

] dx

0

,mr+.+mrr) (a+b,m+mr)

r(a)r(b)q ( (d),mr+.

l'i

-r.(x) = (x/4c /rG*l)oFt(-;c+i

and

tnn)

integrals {2.2.3.3) and {2.2.3.4) become, respectively

r(a)I(b)s\((d),*t*.*mn) zl lCa.ul r(a+b) t,,r,

(2.2.3.7)

(h("') ),m,,)mlm, I . .mr!

These

I(a)r(b)r rTa.5)-l (a+b,m+mr+.*mn) ( (g),*t*.+mn) (f,m) ( (h),mr)

45

Special Cases

$ec.2.2.41

(d+

1

,fi

Q/13c/2r1a*c/21

n! I(c+1) l(a+b+c/2)

{2.2.4.3) replace the Legendre polynomial by the corresponding hypergeometric function[Erd6lyi et aI- (1953) Vo1. II page 180], again uslng (2.2.1. 1) i-n a sli-ghtly different form' We then have Now

I

I tt*) P*(1-2r +2rx) dx /*t- 1r -*; 0 (d+I,!)r(_a)r(b).?,?;1f -.,1,-nib,-n,n+1 l=,.). -- nl f(a+b) 'l:l;1'a+b:d+I; I

lt.t.r.i) (cont inued)

(2.2 .4 .4)

s (2.2.2.3) to (2.2.2.5) may be special ised so that integrals involving the confluent h)?ergeometric function and the Appe1l are evaluated. This gives the formulae

The result

*mn) (a,m*ml+ : +mk) (b,m1i*1* . +mn) (c,m)

b-11$

4(r 1

-'

b-1,,

-f(a)r(b). r(a+b)

(3)

a,-(, -*) /*tt

[ch.2

Integrals ofEuler TYPe

i*u-1(r-*)b '0,.I

F,,

(c ; f ; sx) F, (d, e, e' ; g,

1r-*;o-1 r.lt,^) rll rrr*l . . rlkr,u*)a* )*"-' 0lK

g' ;rx, tx) dx

. r r a : :-:d:-;c;e;e' i.ls'r't) 'a+b,;-;:; ;;;;;;'

(2.2.4.s)

'

I,p. (c; f; sx)F3(d,d r,e,e I ;g;rx,t)dx

I

o-'

b-1

v(.,s*2)

F

(r Ir -x],t)dx.

(2 .2 .4 .7)

(2 .2 .4 .8)

133 and

Carlson

(1961)

If ue make a quadraLi,: cirarrge ui Li,e valiaLle of integration, z=xt, (2.2.4.8) assumes a standard form and it may be evaluated AS

rCei#llrCul ,,,f ' z

.I,..^a+c+l.1.]. I ,'.-,Z,bi-:c, 2 ;2i7;_r,r,rtl

rti)l

2ctfa*!*l ''t -er\ 2 *b)

lr*.*r L-;a+b::-:; = :-:c+1

'

(2.2.4.13)

:

r(x,b,m) = *rjsl +,i,+,t;|;*2,^2* ',-o*').

the two0special functions of the i-ntegrand may be expressed in hyoergeometric fo:m by means of the forrnulae y(a,x)= ..a (2.2.4.9) x* a-l- 1F,[a;a*1;-x) ) )) and F(x,k) = xFr(l/2,1/2,I/2;3/2;x',k-x-), (2.2.4.1O) Erd6l)'i et al. (1953) \/ol. lI p:;e r cspectively.

t.,r,,.,r.1 ,

consider a finite Laplace transform of the incomplete integral of the thj-rd kind.This last functi.on may be expressed as a Lauricella function of thiee variables by means of the formula ICarlson(1961) ]

We

"'^'-"1 ;-;-; '12.2.a.t1

t2.2.4.t4)

then have

l_ r ?-l--b-1" e-SX-.2 ,.[/x,c,ni 1-x.) Jx {,

ox

0

/ 2 1 *; b- I - sx, ='y *^-1 1 -

Now

Sec

r (a) I (b)

elliptical

evaluate an integral lnvolving the incomplete gamma function and the incomplete elliptical integral of the second. This is 1_

,mU)

Now we

We now

i*u-' (t-*)

) . . (dt*1

Re(a), Re(b) > O.

rt, (c ; f ; sx) Fo (d, e ; g, g' ; rx, t [1 -x] ) dx 0 - ::a:d,e:-;c;-ii. - tr I - f(a)f(b).(3)r ta*b::-: - :-;t;g;rr.srr'L'i -r{rrTf l{e(a), Re(b) > 0. i r -*;

+1 ,m,

a :-n ;-mt ;"i-mk -1:1r ^ "l:11-a*b:c+l;dr+l!..idk*l;

(2.2.4 .6)

and

/*o-

(c+l,n) (d, =

r(a)r(b)-(3).-t: : : a :-:-;c;d,e;d',e';s,r,t, - ll-;uf::a+b:gi-;f; -; - , I

47

Special Cases

Sr,c.2.2.41

"

0

(

3)

ri,i,+, t,tr, *,.

2x,

-. * ;

This is .a special case of (2.2.3.2) whlch yields , .",\?ri;l;"' hypergeometric. function of four variables of hlgher order +mo) (r/ 2,nr+nr+nl (1/2,nr) (l/ 2,nr) t ( a+l /2)r rb)

Ti".bitrf'L"(u*1/2,mr+.

x (-r)'t

,,2*s 1-"1'4,

(2.2.4.16)

Re(a) > -l/2, Re(b) > o,

function (2.2.4.16) is related to the hypergeometric functicns studied by the author. See Exton (1972b) and (1973a) '

The

Finally, we give an integral of Euler type which includes a Lauricella function rj") l" its integrand. Thi.s is obtaineJ by specialising Q.2.3.4). Take D=2, E=G=O and H=I, when we have b- i (d'd2 ;hr, .,hn ; 11x, ., rkx, (c /*u-' (, -*) lFr ; r; rxlr[n) 1

Re(a+c) > -1, Re(b) ,

O.

This section is concluded by giving certain cases of the integrals under consideration where products of several hypergeometric functions of one variable and the Lauricella functions are involved. Srrch results as (2 .2 .3 .5) , (2 .2. 3 . 6) and (2 .2 .3 .7) may be used. For example, we have

0

=

r (a) a)l(b)r I (b)

7*k\ t ll*5)-4

Re(a),Re(b)>0. x(c,m) 1fr'mr).. (hn,mn) t"lr...rln.

(2.2.4.17)

Integrals of Euler Type

48

lch.2

2.3 EULER INTEGRAI,S ASSOCIATED WITH THE GAUSS FT]NCTION

In this section Euler integrals involving a Gauss function as well as another hypergeometric function are discussed. Such integrals ma1' be written. in the form I o - t, (2.s.r) o, (c, d; f; .*k; f 1*; d*, |*'- i, -*1

@1

(2.3.2) I^nn n=0"01*'*n-'(,-*)o-'rFr(c,d;f;r*k;d*. Suppose that k=l when the preceding integral is of the same form as (2.1. I.5) and, on evaluation, this integral becomes f(a+n)f(b) r rc,d,a+n;.\ (2.3.3) T(aaS*l 3' 2\f ,r*b*n ;' i' flence, (2.3.1) takes the forn f(a)r(b) " (a,n)h, -,c,d,a+r); \ (2 .3 .4) |("*II;!. (u.b,n) s'2tf ,r,lr*n;'r' provided that Re(a) and Re(b) are both positive. If the rF, and series not terminating, are it is sufficient that l.i . l. ,F, If s=l, then we nust have the condition Re(f-c-d) > 0 as xel1, If either c or d is a negative integer, then we need only retain the condition upon a and t'for the convergerrce of (2.3.4).

I-et us nor. suppose that c = -N, a negative integer, and that b = d-f-N+1. Saalschlitz's theorem [Slater (1966) appendix III] may now be used to sum the Clausen function of (2.3.4) if s=I a1so. This sum is

(f-d, N) (f-a, N) (1+a.-f ,n) (1+a+d-f -N,n) (2.3. s) (f ,N) (f-a-d,N) (1+a-f-N,n) (1+a+d-f ,n) The expression (2.3.4) now takes on a more elegent form, that is (',') (1+a-f,n)hn I (a) r (d-f-N+1) (f-d,N) (f-a,N) ; (2' 3'6)

ffi

"lo

G*..-*,,) G.ra-r,"r'

If we 1et k=2, then (2.3.I) becomes fc,d'a/2+n'a/2+r/2+n:-l t(a)t(b) T (r,r,)

+ffi.io6ffixFslr,+*n,S*n ''-]

Q'37)

tric Function

49

If b=1, then the previous 4Fj serles reduces to 3Fr(c,d,af 2+r,;f ,a/2+I+n;s)

(2.S.g)

and on letting a = -N and f = d-N, SaalschUtz's theorem may be applied once again if s=1. The function (2.3.8) may then be writ ren

where R.e(a) and Re(b) > O, and

k is a positive integer..ds in the previous section, it is assumed that f(x) can be expanded as a power serles of the form (2.1.2) whose radius of convergence is not less than unity..The integtal (2.3.1) may thus be integrated term-by-term and may thus be written

Generalised Hypergeome

Sec. 2.3. I I

(a,i

2+l-d,iJ) (L,N) (al2+]-d+N,n) (a/2+t ,n)

(a/ 2+1, N) (1 -d, N) (a/ 2+r-d,n) (a/2+

[2. 3. e)

n)

1+111,

Hence, (2.3,\) is now equal to * (a/2+1-d,N) (i,N) ! (a,n) (a/2+1-d+N,n) (a/2+I,nJh n a(a/ 2+),, N) ( 1 -d, N) !6G* t,n) (a / 2+7-d, n) (a,/ 2+ 1 +N, n) n

(2.3. 10)

(2.3.4), (2.3.7), (2.3.6) and (2.3.1O), perticularly the two latter, readily lend themselves to the evaluatlon of integrals invoiving the Gauss function. The expressions

2.f

.l

Generalised Hypergeometri.c Function

The integrals rrhose generaJ form is 1

. c-l

h I

L

1, r

r

t.

J^*''fr-..i)"'rFr(c.d;f;sx")aF11((e);(hy;rx^)dr 0.and

I - ,

r.

Jx-'(l-.r)"',Fr(c,d;f;sx")cFHf(C);(h);rIt-x]")dx

(2.3.1.I) (2.3.t.2)

otr-

k is a positive integer, are noi\i inrrestigated. It j,s assumed that, for convergence. Re(a) and Re(b) are both positive.Also, unless the generalised hypergeonetric function is terminating,we take it that G < H+1. where

If the inner generalised hypergeometric func.tion of (2.3.1.i) a series in its argunent, this integral becomes i f fel , n; rn i. a*kn- 11r-*1b-1rnt(c'd;f ;s^k)d* ' (2'3'1

expanded as

,!oITD;;l"r;" The

inner integral of the previous expression

r(a+kn)r(b) , rJ rTa;hh)- z*t"t*klf.,d,eiE, a*blkn*k_i .

k "'- k L"^*uln .

,

3)

be written

,

(2.3.r.4)

;.,:l

(2.3.1.s)

;J

I

so that (2. 3.1 .3) becomes r(a)r(b)-k: z;cfa/r. , , Ia+k-11/k :c,d; (g)

-r (a*U)-tr: r ;H [a*b] /k, |

may

IS

Ia+b+k-t]/k: f;(h)

Integrals of Euler

5o

Sinrilarly, the integtal (2.3.1,2)

Type

[Ch'

may be shown

2

to be equal to

k is a positive integer, It is understood that Re(a) and Re(b) are both positive and that al1 the series are eithel'convergent over the range of integration or terminating. Other similar integrals also exist which may be dealt with in the same way as will be used in evaluating (2.3.2.1j to (2.3.2.3).

A,trmber of simprer forms of tlese two Iast resul't, *tr!';]'l;l]t' cd, and we givc the following example: L(:1 ir = -N, b=c-d-N+1 and k=s=1. The inner integral of (2,3.1.5) rl:ry then be expressed as a Saalcchutzian Clausen series of unit il rgument, which when surnmed takes the form

0n expanding the Kamp6 de Feriet function and interchanging the operations of integratj-on and summation, (2.3,2.1) becomes

a , I+a-d,(8);", E ( *2'H*2\l *a*c-d, l+a-d-N, (h) ;' I' (2 .3. 1 .8) 'l'hc generalised hypergeometric function assoiiated with (2.3 ' i . B) rilay be summed in many cases for special values of its parameters and variable. Suppose that G=2, H=1, g.,=i+a-d-N,gr=1+a+q-d "r1 h=a. The binomial theorem may be appli6d to expreis the resulting form, and we obtain the expression ,Irn series in closed l(a)r(c-d-N+I)(d-c,N)(d-a,N)

+a r

c-d;a;rx)

(,-*)o-rrr, (.id;.*k;d*. (2.3.2.4)

The inncr irrtegral may be uritte,r

llcnce, (2.3.I.2) nay now be written

I , ^;\r - "2Fi (-N, c;d; x),F, ( I+a-d-N,I .l^t'-'( I -x)'

*a+km+kn-,

I

(2.s.1.7)

.

51

Double l{ypergeometric Function

2.3.2]

where

I I I.(;r-)-1.{!l-o:t

2.3.2 Double Hypergeometric Function

'l'itree Eulcr integrals, eaCh of which involves a combination of Gauss function and a Kamp6 de F6riet function are

a

)*o-'1r-*;b-1rr, {.,d; r;rxk) rF;t;il: , [i] ; [],[:];,*k,.*k1d*,

(2.3.2.1)

I

[, '- ' (, -*) 0

l

I

"' t, 0

1l

o

t

rF,

(c, d;

f; rx

-*)b-1rF, (c,d;f ;rx

.v 'sx",t)dx (2.3.2.2) :r

1

r

-*1k,.*k;4*, (2 .3 .2 .3)

i.

+r

(2.3.2.7)

and

,

c

/k: : - : (p) c,d;b/k,., [b+k- 1l /k r; tqj

:a/k,., Ia+k-1]/k;

, Ia+b+k-1]

that k=r=l, c =-N and b = d-f-N+l, then, by Saalschtltzts theorem, we may express (2.3.2.5) in closed form:Suppose

r(a)r(d-f-N+1) (f-d,N) (f-a.N) f (a+d-f

-N+1

)

(f

,

N)

(f -d-a,

(a,m+n)

(1+a--f,m+n)

N) ( 1+a-f -N,m+n) ( 1+a+d-f ,rn+n)

(? \ 2

q)

A special case of (2.3.2.1) may now Lre evaluated as the more compact expression which follorvs: r (a) I (d-f-N+1) (f-d,N) (f-a,N) I (a+d-f-N+l ) (f ,N) (f-d-a,N) : (h); (h') i" ,.,, ." "G+2:B;B'.(g),a,1+a-f 'P+2:Q;Q''(p),1+a-f-N, l+a+d-f: (q); (qi) 'J' r/ '

Similar results follow from

(2

.3.2.7) and

(2 .3 .2 . 1O) (2

.3.2.8)

.

52

Integrals of Euler Type

2.3.3

lCh.2

giv e ab rief I

gi ven

0

abov e may be generalj,sed of the i nt egra I

d i scu ssio n

- b-1 '(r- xJ2

.A Jx

F, (c ,i;f

k. -G: H ;1x lhP.Q

further and we

stiIl

[e) (t,') ;. ; (h!"], rr. :

I-(( ),*1 +.+m

t,r');.; lrnl' L

(q('J);'

p

The gen eral i sed Kamp6

----- -

)

n-

.t-.t..((q1 ),nr ).. ((q(nl ;"n'In 1

"l

x

a+k mr+.

the i-nner int e gral

I (a+kmr+

is

! (b) +-+km ) i'{a+b+km s 2'Ln' .

+kmr_,- 1

1r-*1b-1rr,

{";di.*k)a*, e.s.3.2)

equal to

L

, +kmr_r,

c, d; a+b+kml+ . +km', , f

;

r)

(2 .3 .3 .3)

Hence, (2.3.3. 1) becones

(a,km+kmr+-*kmrr) ( (c),m1*.*mr.,) ((h'),mI) . . ( (h(n)),mn) r (a) r (b) r' (") r (a+bJ (a*b,km*kmr*. +kmr.,) ( (p),mr* *mr.,) ( (q, ),*t) . . ( (q . ),*,.,) m

m!m-I..rn ln

r(d-f-N+1) (f-d,N) (f-a,N) (a,m, +. +mn) (I+a-f ,mr, .,mn) I (a+d-f-N+1) (f ,N) (f-d-a,N) (1+a-f--N,nr+ *m_) (l+a+d-f,n,,+.+m ) n' L n' and so, 12.3.s.6)

1f-

/*'-l na

1r-*;b-1rr, t-N;d;.r

n

(2.s.s.4)

I

It wj-I1 be seen 16at-, in its gener'al form, the integral (2.3.3.I) cannot be evaluated in terms of the generalised Kamp6 de F6riet functi-on in its compact form. Two results may be deduced from (2.3.3.4) which are more elegent, however. If G=P=O and H=Q+l, we have the formula

k

rF

:(h');.;(h(n));.

;I ii;] : (q') ;. ; (q(")r,t'*'''t"

--

, , )+a-f :(h,);.; (h(n));,c+2:lt [-lg;, ' | (nlr't,','nl "l +a-f-N,I +a+d- F: (q') ; . ; (q'", r, 'rr.S. P+2:Q l_(p) ,l Sf. Z1

Here, the integral is evaluated in terms of a generalised Kamp6 de F6riet function with the same number of variables as that in the integ rand, but with its order augumented by. tro. A number of special c ases of Euler integrals associated with the Gauss function are discussed in tlre next section. 2.3.4 Spccial

Cases

Let us consider the Lulcr integral ol- a 1;roduct cf Legendre poly-

nomia I s

1

h-l r a-1 (2.3.4.'t) lx" '(I x)" ^P (I-2rx)P It-2sx)dx. 0-n which may be written in the form I r a-l-- -b-l ix* ^(t-x)"'rF, (-n,n+I;rx)2Fl(-m,m+1 ; I :sxJdx, (2.3.4.2) 0 where, for convergence Re(a) and Re(b) must both be positive. The integral (2.3.4.2) is clearly a special case of t2,3.1.3), and so it may be evaluated as

;l

r(a)I(b)-k:Q+11-a7k , [a+k-1]/k :c,d,u,.,u; (h') ;.; (t (n)); --TG;Efk' Q [tr*u]/k,., [a+b+k -rl/k: f u,.,u; (q');.; (q(n)); r,s1,.,sJ, Q.3.3.s) _l

where

lrl, 1111,.., lr,-,1 < 1 and Re(a) and Re(b) > o.

o,.

I (a+d-f-N+1) (f , N) (f-d-a,N)

;(h(")); ]*u-'1r-*1b-Irnr[.,.i;r;rxk)r::a;t[-:(h');' 'u:qL,,o,l;:;i;(,);;','"=,,*ldx

o

.:l

_ r(a)r(d-f-N.+i) (f-d,N) (f-a,N)

,

+kmr.,)

r mm_ s,I-.s x---- 1n --F^(a+km.+

that k=l and c = -N, when the Clausen functibn of (2.3.3.3) may be summed by SaalschUtzis theorem if we also put r=1 and b = d.f-N+l. Hence, (2.3.3.S) may be written

----;--i-

0

and

53

r (a)

*k,.,. *il o*

(2.3.3.1) de F6riet function in the int e gran dise xpan ded as a mult iple series, and if this series j-s con erge nt ove rth era nge of ln tegration, the integral (2.3.3.1) tak e 5th e form (( o ) ,mt +,+mn') ( (h' ),mt ) " ( (n(n) ..n.m J,mn)s11..snn as a nex ample.

Special Cases

Now, suppose

Irlultiple Hypergeometric Function

The resu lts

Sec. 2.3.41

(2.s.4.s) Other convergence conditions are unnecessary becarlse al1 +-he hypergeometric functions involved are terminating. If, in addition, r=b=1, a special case of (2.3.1.8) arises, so that (2.3.4.1) now becomes

(-l)n(l-a,n)

;]la-nilJ'2[

-,?-D,m+1,-m;-.

a*n*t, I

;=J

(2.3.4 .4)

[Ch. 2

Integrals of Euler TvPe

54

55

Sec.2.3.41 1

Now

lct a=n+l ani

1=|

when we have

using Caussts theorem

n ! (2n-lrl+

G;ilEn-l rl'wc

now

take a=n+l and

m=2n+1

4t(a+o) 1..a-1 -..b-l x ,, [I-x.) ;-j

the further simplification

I,m)

(2.3 .4. s)

' ,m) (2n+2,m)

in (2.3-4.4), this takes the form

nl (l-r)2n*1 (n+l) (-2n-1,n)

(2.s.4.6)

investigate an Euler integral of a Legendre pol'ynomlal and a Ccgenbauer PolYnomial. This is l, ,-lh-l -^c (2 .3 .4 .7) J x"' (I -*) "-' Pr., (1 -2rx)c|(1 -2sx)dx. '0 Wo replace the Legendre and Gegenbauer polynomials by their.corr"=pondi.tg hypergeometric representations. The formula {2.s.L.3) nay then be applied- It j-s clear that (2-3.4'7) is now equal to

Wc now

nr!f (a)l(b) -l ;2;2, a :-n,n+l;-m,m+2c; "',, r ; c+l/2 'L'J)' (7c,m)TT;t)'1:I;1\a*b:

(2.3.4.8)

special cases of this result are now consi

(1976) page lO8.

(3

.l -2.2)

anci (5 .i.2

'3) respectively

(

(3

,),n)

(c),m*Jr) (a,m+p) (I -b,P-nl !9,r,rJ-( (9),m) ( (d'

:

L2 .7)

),n) (rz)m(-s)n21.

m,

0 Z

If jntegrals involving hYPer geome tric functions of several variables are considered, the n umbe r of possibilities increases very rapidly with the number of i ndep e ndent vari.ables of the functions concerned. We now discuss the fo 1 iowing two integrals which lend themselves most readilY to c onve n ient treatment: _i ( n).. (d (d') .; (3'l'3'I) 7,,r-, rr_rrb-r ' e!:n[tc): Y'ul-(r)'(e') .;G ( ,r],'," '''rJ du i and

1 a-t.[c) : (d') Ju (l-uj-b-1 -C:D ''F,G 0 l(tt , rr')

:::i:ll ''.,

r.

l(

.ji,,,,

,rku,

,,,,,-,1_l du.

(3.

]. 3.2)

a.n functio" al,I is the generalised Kamp6 de Fdriet function of Karlsson (1973), and it is assumed that Re(a) and Re[b) are both positive and that lzl is sufficlently sma11 to ensure that the hypergeometric functions in the integrands are convergent over each range of integration. If C+D > F+G+1, then these functions must terminate, and if C+D 1 F+G, the integrand in question xi11 al.rvays converge. l^Jhen C+D = F+G+1, each case must be conside::ed The

separately.

If the inner generalised Kampd de Fdriet function of (3.1.3.i) is expanded as a multj-ple series, term-by-tern integration is allowable if the above .inr".g".,." conditions aremet andthis integral ((c),ml*.*m,.,) ((d'),mr) . . ((d(")),mn)r'[t. .rmn

(r-u)b,,:;B;B:,[;] ; [i] ; [g:] ;,",su)du

a

(a+m-+.+rn .b-'l '

.

n' zI lm- l. -m ((f),mr+.+m ) ((g'),*1) .((e(")),* n'ln (3. I . 3.3) As before, the incomplete beta function is expressed as a Gauss function and (3.1.3.1) may be written as (a,m*T1*.*mrr) ((c),mI*.+mr) (1-b,m) ((d'),11) . . ((d(n)),mn)

,llrBl'

the following example of a definite integral of Laplace type which involves a K-amp6 de F6riet function:-

we also have

7u,-,

Multiple Hypergeometric Function

I

.

and 1

3. 1.3

59

becomes

Similarly, the integrals take the form a .-Z- F (5J ('a

,za /

Multiple Hypergeome tric Fuuction

Sec.3.1.31

(."-,-l5*.

*mn)

((f),m1*.*mr) ((8'),mr)..((g(n)),mrr)m!m,

! - -mrr!

(5.1.3.4) (rrz)ml. . (rrz)mn. If we investigate (3.1.3.2), we finC that it is equal to (a,m+mr+. *mr.,) ( 1 -b,m-m1+l - . . -*r) (b,m1* I + . +mn) ( (c) ,mr+ ' +ntrr) 1

(3.t .2.e)

a

A few special cases will be discussed in Sectj'on 3'1'4

*(

(d'

),*r)

. . ( (a

(n)

),m,.,)

zm(rrzlnr

..

1ruzlmo,-'u.rr:1..

i l;

l;ir'"

(r()

Ich.

Definite Integrals and Repeated Integrals

3

Nr:xt, tltcr two similar definite integ.rals of Laplace type which I'ol low are cvaluated:-

,r.u,.,r-J o, "" te(n)),''"" """J

|..,r-r ^-u .c:Dl1c): ( d');. , (d:']) ''Ftt[rtl:(g');.,

'nt' " a

7,f

(a+m+mr+.*m ) ( (c),ml*.*mr)

(1d'),mr)

..

r;.-r-,P i,u-t,,-,)b-1Ll(ru)du =

the integral

(g(') ),*

lt

-=

_l

-C:D tF,

G

l-t.l,ta,);.;(a(n)); I'''k'''k*l""lo" l-irr,1r');.;(e(n)1r

(a,m+mr+. + mU) ((c),mr*.*m,.,)

m.m, . (rt.)'-'k , .m, (r.z)"'1. mlmr!.....rr!

z"'

(3-1.4.4)

Aiso, using (3.1 .2.8) , we have 1 .-t h-r Ju*'(t-r)"'Fa(c,d;f,f'

;ru,sIl-ul)du

0

nl,

11+l k+I' . .rnn

(5. 1 .4. s)

m

(3. I .3.6)

3. I .4 Special Cases

certain represenl_ative exailples of special cases of some of the integrals tf Sections I.1.1.to 3.1.5 will now be discussed. If we the Gegenbauer po11'nomial in the integrand of "*pr".. (3.1 .4. 1) Jo'-1 1t-,r;b-1 cl it-z.r)a,, 0 as a Gauss function, a special case of (3.1.1,2) is obtairred, and so we may apply (3.1.1.5). Hence, the integral (3'1'4'1) may be

l{hen turnlng to special cases of definite integrals of nultiple hypergeometric functions, apart from the product of several single hypergeometric functions, very few classical speci.al functions falI directly into the'mu1tip1e hypergeometric category; even though such functlons appear in divers applicatl"ons, '-he1' have only recently been presented in hypergeometric form. See Exton (1976) Chapters 7 and 8, for example. B.C. Carlson (1961) has fairly recently expressed the incomplete elliptical integral of the third kind as a Lauricella function of the fourth kind in

three variables. It thrrs follows that the integral zo-' n 1rlr, s,k)du /,.,'-l 1t-u;

(2c,n)zarL:2;I, a:-n,rr+2c;l-b;

=i_l.i_+i'iio('I' '

- -\ ta (s'r'4'2) ..iit'"''- -"'z't'z)'

we have the following integral invoi.ving a generalised Lngucrre poll,nomiaI : -

(3. t .4 .6)

0

be expressed as a special case of (5.1.3.4) so that it is equal to 'I 2 (a+1/2,m+mr+mrtmr) ( 1/2,m-r+mr+ma) ( I -b,m) (i, m, ) (i,mr) ( 1, mr) a+I/ TZT ^,1t1 2 L {a+3/2,m+ml+m2+m.) (5/2,mr+mr+ma} a+l/

may

1

^w

.. r^ 1r2 11^ 7 1r2 12 11* 21-kr2

-)

evaluated t,o give

!t

rzz ,(3)l. a ::1/,2:.-:.-;.1 /2;.1 ;.1-b;.u2r,-rz,z) (a*I:: I :-:-; - ;-; - ;o 2a t

((d'),rn1)'' ((d(n)),mn)

In both (3.1.3.51 and (3.1.3.6), it is taken that Re(a) > O and that ]zl is sufficiently smal1 to ensure the convergence of the multiple hypergeometric functions of the integrand over the range of integration, unless these functions terminate'

Sirnilarly,

du

by means of (3. I.2.6)

I

n.l

1 a-l -u Ju " 0

z/ru-' (, -,r)b-1 r(ru,k/u) 0

z*[rrr1ml. . (rrz)mn mlm, 1...rn.,

.

;

elliptical integral of the third kind rnay be written as an Appe11 function of the first kind. Hence, we may evaluate

((d(n)),mn)

*rnn) ( (f) ,m1*. *mr) ( (g') ,m1) . . (

"rl i ;i(^:, ;.;l',t -b'.,,,,)(3.1.4.3)

The complete

7"

;"1'(a+I,m+mr+.

6l

Special Cases

Sec. 3.1 .41

If

z)

m3

(3. i .4.7)

take C=D=F=l and G=0, C=D=G=l and F=0 and C=D=O respectivein (3.1.3.5), for example, we have Z ir[') q.,dr, .,dn;f ;r,u, .,rku,rk*r Ir -u], .,rn [] -ul )du = fu '-' (, -r)o we

l

1y

0

(3. 1.4. (cont inued)

B)

Definite Integrals and Repeated

62

(cont i nucd) a (a+m+mr+. +mU)

Zt

(drmr)

i,,'-'

lntegrals

(l -b, m-m11] - . -*n) (b,*k* l*.

[Ch.3

(d,,,*-) zm6rrzl*1. . (r*zlmk(-r**r)*k*1. mlm, !.. .mr., I

,,-,,0-tnl",[c,d,,

. ,dn

;gI,.,8r,itl''.'Iku, .k*l I I-ul, .,rn

3.2.1]

As

beforc, it will be

assumed

tr -,r!

.

(.rr)mn

i3. I .4.8)

(a,m+n+p) (c,n) ,a_ Lt_

a,-t

aL

(a+1,m+n+p)

(1

-b,p) h*zm 6rz1n1-

(a,m*m1 +. +mk) 6I -b,m-m1*1 -

_

.

(b'

),rr

) . . ( (u(nl ),m,.,) zm(rrz;mt . . fruz;mli m!m, I

(-ru*r)mk*l

. . (-.,.,)*n

.:.mn!

3.2 DEFINITE INTEGRAI.S ASSOCIATED WITH TIIE KUMMER AND BESSEL FIJNCTIONS

investi-gate integrals of the following types:z,

a-I-b-l-l.t (c;d;ru)f(u)du, Ju- (1-u)-

(3.2.r)

:r 2,-L *rFr(c;d;ru)f(u)du, Jr- ^ " -l.l -

(3.2.2)

'!u"-'(r-r)b-l J. (ru) f(u)du

(3.2.3)

0

z/ua-r e-u J.(ru)f(u)du. 0

(3.2.6)

n)

n Ip

t3.2.7)

!

1 -b, p) hr r^ (- 12 12 / (--1\L (a+c)f (c+i)L(a+c+1,m+2n+p)(1+c,n) n!pl

and

,r-*, (r/ 2)c .

(a+i)Tlff) t

4)'( - r) P

(ar c,m+2n+p)hrr'( - -212 / d)n(-r)P

(3.2.8)

(3.2.e)

specialisation of f(u) in the forrn of a single series in its argument is the generalised hypergeonetri.c function ,Fa, see (3.1.1.1). We are thus led to consider the integral 1a-t-, .h-l (3.2. 1 . 1) /"'-'(r-u)"' rFr(c;d;ru)FFct(f) ; (s) ;su)du, 0 which is a special case of t3.2.6). Hence, (3.2.1.1) may be written as

The most obvious

(f) ;c;I-b; -za -(3), d ; i-:-:-; (3.2.t.2) (a+I: : -;-t-t(g);d; - ;t''t:z'z)' ;*" Let f(u) take the form of a Kanp6 de F6riet function, so that, if applied to (3.2.2), for example, we have

f,u-1 "-',F, (c;d;ru).!::::l rff]: If]: i9i] i,u,.,10,, 0

0

and

p

3.2.1 Generalised Hypergeometric Functions of One, Two and Several Variables

(3. 1 .4. 10)

We now

s

-*r,) (b,m**r+ ., mn)

((g'),mr)..((g(n)),rn) (

z1

(a+c. m+2n+p) ( a+c" (r/2)" r/ t) t z*

(3. r .4. e)

and

,a. aL

(3.2.s)

a (a,m+ n+p) (c,n)h zn(rz)'(-r)P (a+ 1 , rn+n+p) (d,

'luo-' rL-u)b-1DFG((d, ) ; (gi) ;rrrr) ..nFc( ra(k)), (e(kl) ;rru) o,ora( (k*I), .,r(k*1) (n)', (d ; (g(t] ) ; rn { I -ul )du ) ;11*1 [1-ul ) . . nFc( {a

expansion

(d,n) nlpl

Z a'

!...mn!

that the series

63

has a radius of converg.ence which is not less tharr I z | . With thi assumption, each of the above integrais may l-e evaluated using term-by-term integration, so that (3.2.1) to (3.2.4) become, respect i vely

^1.

m!m,

.

Functions

f(u)=I ' .r!ot-rn n

*mn) (c,mr+. +mr) (a, m+m, +. +mU) (1 -b,m-mU *1-, -mn) (b,m1*1*.

a

Generalised Hypergeometric

tmn) (c,ml+, +mn)

--'ffi,* --J-

.

Sec.

_ -alz\

(a,m+n+p+q) ( (f),m+n) ( (g),m) ( (g'

),1) (c, p) (sz)m (tz)n(rl:)Pza

(3.2.4) Similar:1y, integrals i.nvolving generalised

Kamp€

(3.2.i.31 de F6riet

Definite Integrals and Repeat Integrals

64

l trnct

resul

ions of t

severa I

vAriables may be evaluated;

lch. rse have the

N

z-2 F:Gl (r):(s,);.;(e(n));, .,2 r /ua-r e-u J.(ru)F 1*'''" ,.,l o'

I (a-b+1 ,N)

za*

. a(1-b,N) (I+a,N)

''*Lrr) :(k');.;(t(n));

0

-

C,+2r

rd,a-b+1+N, (c);.^^. D+?\a-b*I, a+l -)J, (d) ;" )

(3.3.1.2)

3.3.2 Double Hypergeometric Function

(r/2)c r(a+c)/4, ( [a+c] / 2,p+m+nr+. *m,,,,) ( (f),mr+. **ar) - - (a+clilc+ll-z ([a+c+2fl2,p+m+mr+. +mr) (c+1,m) ((jJ,mI+.+mn)

consider the two integrals given below, eacl'r oi- which involves a Gauss function and a Kamp6 de F6riet function. We now

,.mn (s'),ml) . . ((e("),*; {-r)p/ 2 (-r2/r/ q^ (trlz)nt ..lt--n {z) 1ik'),mr)..((t(")1,mr.,) plm!mrl...m (3.2.t.s)

]

( X

65

Double Hypergeometric Function

Sec.3.3.21

(3.3.2.r)

ru, su) du

!

It appears that no general reducible forms of the integrals discussed in this section exist. 3.3 DEFINITE INTEGRALS ASSOCIATED WITH THE GAUSS FUNCTION

A number of fairly compact results mav be obtained if integrals of the general type Z

a- I 7 j r"

ZF

r

( -N , o -N

;b;u/

z)

f t u) du

(3. 3. 1 )

On expanding the Kamp6 de F6::iet function of the integrand of (3.3.2.1), the conditions of convergence imposed upon (3.3,2) enable us to integrate term-by-term- A double series of 3F2(1) functlons results which may b,e reduced by Saalschiitzrs theorem. Hence, (3.3.2'1) may' be written as

:(d);(d' ,'N.1^-b'1,N) F*2:G;G' \(f),a-b*l,a*[+N: [g) ; (g' ;O-E,Nfia-l,N)' "C+2:D;D'ric),a,a-b+i+N

'

t1

0

\r

are considered, As above, it is_taken that

r(u) =

,,ionnrn

Simrlarly', G.3.2.2) takes t-he form (3. r. 2l

convergent for lul :1.1, and N is a non-negative rnteger. Hence, (3.5.1) may be written as a sum of Clausen functions. The appli-cation of Saalchutzt's theorem then enables us to Put the integral (3.3. I) in the form (a'n) (a-b+1+N'n) hnzn (1,N) (a-b+1,N) zaI (3'3'3)

(l-b,N)(a*l,Nfi-Iffi'

This last result may be employed to i.nvestigate a number of integrals involving the Gauss function together with other hypergeometric functjons. 3.3.

I

Generalised Hypergeometric Function

If Re(a) is positive, a Z-

number

of integrals of the form

(3.3.1.1) Ju" zFt(-N,b;b-N;u/z)aFr((c);(d) ;ru)du 0 are of frequent occurrence. Here, we have specialised (3.3,1) by taking f(u) to be a generalised hypergeometric function of one variable. The formula (3.3.3J may be applied, so that the integral (3.3.1.1) may be evaluated in the form .

A-l

z a 7\

. J . L . J )

,uN:ir-brl,N) rC:D+2;D'r(c):a,a-b+l+N, (,r) :(d' ;(t-b,NlG-l,Nt' F :G+2;c''(f) :a-b+1,a+I+N' (g) ; (e'

(3.3.2.4)

3.3.3 Multiple Hypergeometric Function

results may be generalised to a further degree by lntroducing a generalised Kamp6 de F6riet function of several variables into the integrand of the integral under consideration. We give the foltowing two results as examples which may he obtained in exactly the same fashion as the expressions discussed in the previous section. ; rb(n)'t' u' ,r)rF,GL,ar 7,u-' rr r(-N,b;b-N;u/z)r::P l(.^]',,r,),., !o:]'' (e,rrr,.,", The previous

''"']

o

,'Nl 1u-b*1,N; oc+z:o a(1-b,N) (a+1,N) n*2:c

[(c) [_(f)

,a,a-b+1+N

;(d');.; {a(n),. (g');'; (g(n) ) ,

,a-b+1,a+1+ N;

-1

11z''''rn

.t "l

(3.3.3.1)

66

Definite Integrals and Repeated

Integrals

[Ch.3

(3,3.4.4)

and

z, (b');.; (b(n));- .. - -l o" -. -rF, .lr*^-tt (-N,b;b-N;t t/z\eg,?[(.): '/z)FF"eLi.,, i*';;., i*(,),,"''12" 'r1uu

,.

b =

,uNI

6u-b*1,N) .c:l*zf-1c;:a,a-b+1+N , (d');P,q, (d") i.-;

;fi5,Nl-(a+l;l0'n:c*2fir)

P'q' (d(n)''

(n) 'r'r'

'"' ' ''C ' (r i - ; ' 2)

p and q are dummy parameters i-ntroduced to retain tflo ilotation of the generalised Kamp6 de F6riet function on the ri.ght cf

the

tou- 1 -N'

0

zN(1+N)

TN;I;ITN;f (

t

-2u/

(3.3.4.6)

'

z)erf (ru) du

15.5.4./)

form

(3.3.4'B) l*)i"',F, 'rjov\ 1\r 'c a , [-N,b;b-N;u/z) rflQ/2;3/2;-r212;dr.

In this case, it is most convenient to refer directly to (3.3.3), (3.5.4.8) may be written as

The s.pecial Gauss function

2(a-b+2,

2Fl

(-N,b;b-N;x),

(3.3.4;1)

wlrich occurs in the various integrals in Sections 3.3 to 3.3.3' may be expressed in terms of a Jacobi polynomial

,,. pP-1-N'oit-z*;, 'N (b-\,N) *,j, ,,

(3.3.4.2)

Erd6lyi et aI. (1953) VoI. I page Bl, and its special

case

the Legendre polynomial

P*(1-2x),

(3. 5 .4. 3)

\)

(b-N.

lilluru1a*z,lr)

Thls, in turn,

N) 1

,'*l i

a.

may

r

1a*t ,2n) (a-b+l+\,.2n)l 1/ 2,n)

;, [email protected] il7

; y a;7.N, n

(-1)N(a+N,t'l)za 'lu^-, ,rN\r-;-;rrr,-lru.)ou L), - -G;i;l}a-a"3ta_N,a+]+N, ,F-f ,3t-n:n:1 t;L ,r-?..,p rr-.-,,\r,, = ,i.rr), ,0" G.3.4.4) where Re(a) , O.

that a=r=1,

when

the integral

' ,u., /r*tr-2"/r)P,.,(1-2u)d

I t sZ,

(-r:)2

"I,

I

(5' 3.4 . e)

be expressed as

(b-N,i't)24*1, --fr:ElU ia.:Tf(a*tI s'a l+,+,OgY,5gry | ,iu*z'9_!11'e.r.3 1 '-2'l'22

using Legendre's duplication fcrmula for the Erd6lyi et al. (1955) Vo1, I page 4.

gamma

I

^^

/r,'-'

R*1t -2u/ z)n

(ru, /[su] )du

I I

I, .l

;

(3. 3.4.10) funct io n. See

Examples of special cases of integrals invplving double metrlc functions may be fu::r,ished by the follovring: Z.

have

zn

2(a-b+2,.\-)

b = 1rJ+}.

The generalised hypergeometric function associated with (3.3.i.i) may be expressed as a vari-ety of special functions. l{e consider, as examples, an integral involving a product of Legendre polynomials and an integral involvi-ng the proCuct of a Jacobi polynoniial and an error functj-on' From (3.3.I.1) and (3.3.I.2), we

may

_ -

r

.

Suppose now

-2u)du

is now investigated. Replace the special functions of the integrand by their hypergeometric representations, when (3.3.4. 7) takes

when

when

function of unit argument to which

0

3.3.4 Special Cases

see

z) Pn (i

f,.,^-

where

(3.3.3.2)

Clausen

z

/e*{t -zrl The integral

-l

,'

a

becomes

SaalschUtzts theorem may be applied. Hence,

:a_b+1,a+1+N, (e, ) ;p,q, (f,,) ; . _ .

p, q, (g

67

Special Cases

Sep.3.3.41

hypergeo(3. 3.4. 1i)

0

', a-l - - .. /z)F oG,d; f, f' :ru, su)du, Iu" ' zF ' t(-N,b;b-N;u (3.3.4.12) o where II(x,y) is a complete elliptical integral of the third kind and F4 rs an Appeltr functj-on of the fourth kind. The integral (3.3.4.11) may be expressed as a >pecial case of the standard . form [5.3.2.1), and so may be evaluated to give and

(3.3.4.13) l{e may also use (3.3.2.3) to show that (3."\.4.12) is ctlttrtl to

(3.3.4.s)

be expressed in closed form. The right-hand nember of

zaNI(ir-l>"1,N),-4:0;O,rr,:t-ltrl,N,c,tl:" l' ,-Nl ;,t;l',, l1rl,;1 rlrt.,l

.fifi ,ljfr;l

'r'ir';r'7'\7)'

(,i.3.,1 .14)

t tch.3

Dclinite Integrals and Repeated lntegrals

66

If wo put s's-!+I and d=a+N+I in (3'3'4'14)' to glvlt tho cxpresslon

#ffiffi*$

lrr(a,a-b+I+N;f ,f ' ;rz,sz)

this simplifies

then

(3.3.4.16)

'

discussion of three integrals Thll roctlotr ls cotrcluded t'ythea brief g"tter"lised Kamp6 de F6riet functiiiuifri"C ipocial cases ofconsider the integral

iiin-ii-iirii'al variables'

d,, 11,', r ,rir1,, . ,rrru)du, Iu*-' rn, ( -N, b ; b-N ;|) rfn) t", ar, ., i (3.3.4.L7) 0 case of special a is whon it will be seen that this expression (5.3.4.16) integral the Thus (3'3'3'1) ' tho riSht-ltanJ member of

may

be evaluated

as

. (s. 3.4.17) ,oNl1u-b*I,N; .3:1i-c,a,a-b+I+N:d1; ' idni ,-r...r) ;(T-bTI-(a.1,Nf 2,11_u-u*r,a+r+N:f 'J ,;. ifn, ' which are (3'3'5'1) of Also, we have two further special cases worth) of note:-

io"-'r.,

(-N,b;b-N;5ri') [.,dr,

rrru) du

. ,d

(3.3.4.18) ''rt"' ''t"1

a, a-b+1+ ,'N I 6u-b*1,N1 = ;G:E$G+T,NI F1'1F, t ''l t, a-b+ 1 , a+ and

u^-' ro,( -N, b ; b.-N r\) rr r(. 'o')t '!

=

1,

d

1

;f

rru)'' ,;

2F1

(tn, d,

;

;' icn'dn t'

i:T {"r-,f;1iT)niF2:2fa'a-b+1+N''r'-0, d\r-u,.r., \ui 4 'tt) ''L fl ;.; l_a_b+I,a+L+N:

tn i

f,

;

rnu) du

I r',' '"11

This last result leads to the following integral of several Legendre PolYnomials : ]u'-'o* {, - rrr r, Pml (l - 2rru) . . . Prr, ( I - 2r-u) du ,'(r-N,N)

i-t1N .2:2 'f .,

-f-1a*t,Nl-

'z:t

a

:-ml,mr*l!.;-mn,mn+I

l_a-r.r,a*l*rv:

r

i.i

In discussing repeated integrals of hypergeonetric functions, first of all consider general integrals of the t)?es zz

r^-'(1-r)b-If1r11ar)n i.r"l.i 00

zz I.b).| ,u-t "-" f 1r1(dr)t. 00 We expand the function f(z) in the form

product of

we

(3.4.1) (3.4.2)

and

6

(3.4.3) t(z) = I tl-r* , m=U where the radius of convergence of (3.4.3) is sufficierrtly large for the intcgrals (3.4.1) and (3.4.2) to be treated using term-byterm integration. lre may thus write (3.4.1) and (3.4.2) in the forms

@zz-

I n^i. 6).iru*^-

m=(l 0

.M=UUU

;

^^i

t

(,-.)o-'(ar)n

(3.4 .4)

u

.

b) .1,u*^-1

u-'

td,1n

respectively. The Lnner lntegral of (3.4.4) ' ; (1-b,r)ru***'*'-1 r=O rl (a+rn+r,nJ which may be expressed in the forrn a+m+n-

'

(3.5 .4 . ls)

a

3.4,REPEATED INTEGRAI,S

and

-t

69

Repeated Integrals

Sec. 3.41

,

may be

.

written

(a+m

'

I -b; a+m+n ; z) '

Hence, the integral (3.4.1) becomes a+n-1 hm'(a.m+r] (l -b,r) zm*r z m,

I

as

(3 .4

.6)

I

fo.r;l-rt,

_G;I

(3.4. s)

r=O .

(a+n,m+r) rl

(s .4 .7)

(3.4 .8)

A similar i-nvestigation of (3.4.2) enables us to write

rrrr,.,rr;l

t,:

,.r.;J,

zz i . r"l .i .a-t "-z r(z) (dz)n 00 a+n-l - m+r @ h (a.m+rlz z ' m' 1 (a.n) ' ^(u*n,m+r)r! ' m,r=u

(3.4.e)

lch.3

Definite Integrals and Repeated Integrals

70

I

3.4.1 Generalised Hypergeometric Function

za+n- I - Fl,l;lr u '!:]il-bi.rz,z). G,"I- 'l:c;o\a*n: (d) ; -

(3.4.1.1)

;

ChD

i .t:l ;-;.r,,,1 ','",(a,n) Fl'l;9( 1:C;O'a+n:(dJ;-i

(3.4.1.2)

.

In the formulae (3.4.I.1) and (3.4.1.2), Re(a) > O,and in (3.4'1.1) Re(b) > 0 a1so. If b=l, a simpler form of (3.4.1.1) occurs. This is 7

7

i.,",.i,^-'cFD((c)

; (d)

;rz) (dz)n =

c*rFD*r,til ;,nrn'.,,).

(3.4.2.3)

3.4.3 Multiple Hypergeometric Function

further generalisations of tire previous results may be obtained by the same methcds as thcse used in the pre'rious sections to evaluated repeated integrals involving multiple hypergeometric functions, in particular the generalised Kamp6 de F6riet function The following t].o examples are given without proof:l'lany

z b

I \-I hp.cl "(s)' t:";..,'.trrr,.,.rrl -c'nkc):(d');. ' (I-=)t -) {ur)n ' '"Lf tl: (s');.; (s'-'); '

(a,m*rnr*.*mr) ( (c),mr*.+m.l (_(d'),mr) . . ( (d(s)),mr) Zla+rt - 1 (a,n) " (a+n,rn+mr+.*rns) ((f),rn1+.+ms) ((g'),mt) . . ((s(t)),*r) (1-b.m) (rrz;m1. . (r.z)ms Y

a+n-1

"6:il

:: (c):

F(J] r- -..(rl. - rc\

.r"l,l .u-l, f'o

(c);(d);rz)(dz)n

a+n- I

--

I

-- z-ta,nJ

z

From (3.4 .9) , we have

zz^1 r il-t t e J'(nJ'i z 00

cont rnued.) a+rr -

In this section, integrals of the types (3'4,1) and (3.4.2) which involve the generalised hypergeometric function of one variable are discuss"a. ff the formula (3.4-8) is used, ic follows ihat 1 a-L .- . b-rI.rr[(c); (d);rz) (dz)n i. trl. J z - (i-z) 00

'tl

Multiple Hypergeometric Function

Sec.3.4.31

and

-

zm

(3.4.3.1)

*l,nt:-mJ

;' ;(a('))'.,-, .,r,2,r,.. rS:?[('):(d') i,,r.1,"',r-,)b-' 'r'c[rr):(s');.;(g(=));l-''''k"''k+ ,' o o

3.4.2 Double I{ypergeometric Function

The expressions (3.4 . B) arrd (3.4 .9) may readi I y be extended to double and multiple series. If we take f(z) to be a Kamp6 de F6riet frrnction, we irave the following results which rnaybe given

I (a,m+mr*.trn*) ((c),m1*.*m") ((d'),mr) . . ((d(s)),ms) (a,n)" (a+rr,m+int*.*rk) ( (l'),mt*.+ms) ((g'),mt).. ((g(t)),*r)

a+n-

in terms of Srivastavats triple hypergeornetric function: rAu:u;D'rtc]: (o); (o'),_.L L .-r r,t?r '{,^-' (dz)n (r-,)o-1 eg ?ill r!:l : f:l : till i,,,sz) \!L) ).ln).12 F:G;Gt'(f)

'tl

__ -n xldz)

,(g)

; (g')

(I -b,m) (r, z)mt. . lrozlmk mlmr!..m*l

'J')

;

'l

. _l-,. '.' ' '-rz , sz , z) (3.4.2 .I)

If b=1, the right-hand the more compact form

member

.*li'

,rt f7 i fJ.i.J.",

of (3.4.3.1), for exantple,

7

1\

assumes

,r ' r"=1 3.4.4 Special

Cases

A number of special cases of repeated integrals each involving a single hypergeometric function are now mentioned. The Bessel function of the f irst ki-nd may be represented as a nF., function. Hence, from (3.4.I.3), we have the result

ir!

,it

[ch.

Definite tntegrals and Repeated Integrals

72

(\c za+c+n-l [a*c a*c*l

z

z

) )

3

-l

i.t"r.iru-' t.(rz)(dz;n = #;fir*lil2F31 2'-T' -"2'/nl' Le:;-n,..-*,c*r;

If

we

z

/. c"t 0

and,

In conclusion, repeated integrals of the Lauricella functions tl'; ;"; r5*)'r."'cvaluatcd uv ,p".ialising thc formul a (3'4 '3'3):

J

13.4.4.r) we have function, BesseI modifjed the consider norr

.i*-' r.(r/z) (dz)n

on putting a = c/2 be obtained.

(3.4.4.

1O)

f (c+1) @+|,n)

0

may

73

Special Cascs

Sec,3.4.41

+

(3.4.4.2) given below case special I, the simPler

Z,

(3.4.4.11)

(3.4.4 .3)

f.rnt.I

00 conflIf we replace the Laguerre polynomial by the corresponding (3'4'1'3) as to so use again we may uent hypergeometric iunction, obtain the formul a - a+n- I a -\ir^. ,-.fl -= (c+l,mJz E r rilrr)(cz)" 'a+n,' c+1 ; z'z !2"-' /.tnl. ffi 00 (3 .4.4.4) cases special There are also the further z ^ ^ z .n = ,a*n ,c+n (3,4.4.5) r'1r21(dz)" fnl. Ir' /. 5;i.-rrtl;'"(rz) UU

nz) '

',

(3.4.4.t2)

- -. -n

z) ldz)

(3.4.4.131

and

and

z,.,roz) (dz)n .'l ,u-' r[*) 1r*.,,d1,.,d,,,;a;r! i.r.r, ' 1. rr 6 '' 6

The repeated irrtegral

zz

i. ttl 'l 'u-' 00'

P,n(l-2rz) (dz)n

(3 .4 .4 .7)

=

is now investigated. The Legendre polynomial is written,in hyperin !"o*.ati. forml so that (3.4-4.7) may be evaluated immediately the form a+n- I

#

and if a = l, we have

rt,('i.},'i' ,"' '

(5.4.4.

8)

n, ,-2 (s .4 .4 .e) r['-"rr -2tz), = v^tr-2,,)(dz)n i.c"l.i ft#*lT OU where the repdated i-ntegral of a Legendre polyncmial is expressed in .terms of. a special Jacobi polynomial ' z

z

_a+n-I

fr-;Uf

,

) _A -rrz) -o1 . . . (I-r*z) -'*

..

(3.4.4 -14)

Generalised Hypergeometric Function

I

Sec. 4.1.1

Suppose

that f(u)

may be expanded f

Chapter 4

(u) =

as a double series

h'"'

*l,.'=o

then (4.1.I) now takes the form

Contour Integrals

(-1)'*t'

4.I POCHHAMMER INTEGRATs

of Euler type, such as are discusseci in chapter 2, by contour integrals, where the path of integrareplaced may be tion is a Pochhammer double-1oop. see,lAJhittaker and itratson (1952) page 255. Such a loop begins from a- point P, sa{, bettveen0and 1, Lniircl.s O then 1 in the positive direction and then encircles direction, returning to the same two points -such again in the negative a contour wilI be denotcd by c unless otherwise the point P. statld. The advantagi of such contour illtegration is that the restrictions on the parameters of EL:ler illtcSr'a1s may be lifted. [)oub1e-loop i.ntegials are also a very porvclful tool in thc investi-gation of hyp"rg"onretric Jif ferential sj'stenrs. See E'rd61yi (f 950) The integra,Is

for

erampl c.

lre begin by considering the gcncral typc of intcgral Ja

r'-t(t-,r)t'-1 f1t,1 du, t(ul = r/

where

h*u

III

(4.1'1) (4 .1 .2)

If the contour. c can be deformed, rr,ithout crossing or enci,rcling any of the singularities of the irltegrand of (4.1'1), so that thc series representation (4.1.2) converges uniformly upcn it, then the integral (4.1.1) may be replaced by r+m-I.. .rt-l ' du' (4'1'3) LIrrrr-1(l-u)' I^r.,r/6

m=u

The above inner integral mdy be evaluated by means of tlre useful formula Isee Exton (1976) Page ]7] ) ltrt ) (4 .1 .4) d, {-r)'-t(.,-t)''-1 la li1-11 1 11 -r' ) I (r+r-')

on the understanding that rt i-t int"rpreted as exp(r log rr), where u is rea1, positive and continuous on the contour. llence,

r+r, .^ ._2 o l, (r.nr)'. Grr)-,, I "T , f (1-r)i'(I-r')f (r+r') *!O (r'r',trr) -

Xi., m,n I

m, fl=O

"(l-')n

(4.1.6)

'

(r,m) (r',n) (r+rr ,m+n)

(4.1.7)

Both of the results (4.1 .5) and (4. 1.7) may easil;, be extended to multiple series and will be the basis of the following study of Pochhammer integrals. 4.1.I Generalised Hypergeometric Function

If we let f(u) in the integraJ. (:.I.l) take the form of a generalised hypergeornetric function of one variable, we have

.^ .,2 Yr)) -if(;F r[1-r) r(1-r - --r+rt (-il

r rr(r' (a i;r+r' (bl;x), (4.1.1.1) A,IL:-r which holds provlded that the ccntour of integration, C, can be deformed so that the hypergeomet::ic function of the integrand is con\rergent upor. it. Any values oi the parameters, either here, or later in the text, which make anr- of the associated ganma functions inlinite are tacjtly excludeil .

m=o

(4.i.1) becomes

12ni12

l(1-r)l(1-r')r(r+rr)

75

(4.1.s)

If we let r=bl, then the hypergecmetric function on the right of (4.1.1.1J assumes a simpler form: nFB(ut,.,aA;bt+r',b2,.,bu;x), and if r'=al-bi' also. a further simplification results R-tFg- [email protected]''

''A;bl' .,bu;x).

(4.1-1.2) :

(4.I.1.3)

For special values of x, a number of integrals of the type (4.1.1.1) may be evaluated in closed form. For example, if x is made equal to unity, and A=1 atrd B=0, the hlpergeonetric function on the right of (4.1.1.1) may be summed using Gauss's summation theorem, see Slater (1966) Appendix llL This gives the result

, b-I.

Jc'

t

It is understood that tle(b+r-a- O. Sirnilarly, it follows that

(b') ; I u 'l'2'B) (g') ; i-,rrr,-*r" bed ed uced from (4.1.2.5) and {4.1.2.5), crmul of a general natirre exist, to

79

representation and again making rrse of (4.1.1.1). If, in addition, r = c+l, .we have the interesting special .case

' (4

Special Cases

Sec. 4.1.31

we take

:I;:;iXi:l)"'*, ."*l du

(b

)'

4

-' II(cu,k/u)du.

rl -1

(4. 1.3.. 10)

This is an integral of a complete elliptical integral of thc third kind, whi-ch, in turn, may be written as an Appetl function Fl. This integral may then be expressed as a special case qf (4.1.2.5) which may then be used to evaluate it. in the form

+= l= !?::'t'2III-rtrGlr')TIr-r)

t;7,r,r/?

:.r/Z;Li12,

(4.1.3. rl) -.) "z:o;o(r+.r',1. -,-, We now give an example pf a double-loop integral involving an Appell function of.the second kind: -1 F Ia r'-'(1-r)r' 2(a,b,b''d,d' ;ux, [1-u]y)du r+rl ) (-f )' '' (2ri)" .\:2;2 , a (4 .t . s.t2) ,2:

ffi'l:l;t\r+i'

}!

t{0

Contodr

Integrals

{Ch' 4

'l'lrcsc arc

)

-(n), ['A'(a,bi,',b

.ri

r+r'

_-l

(4.1.3.13)

I

t'

(z I

ux-

L .^. -(nJ, l'd'la,Dl,.,D n'u'1-u''

D1_ l:1)

'1

[] -r) I' (l-r' ) f (rtrr

^2'. r

)

where Re(a+b) >

' c+1l(a+f) l(a+g) r(b+f) l(b+g) 1 _-_' ",.*'# I i(a+u)r(b+u)r(f-u)r(g-u) O. Consider

Suppose,

(4.2.6)

the integral

4.2 BARNES INTEGRAIS

An jmportant class of contour integrals is that where the contour of integration is the straight 1j-ne, often indented, lying para11el to the imaginary axis, in the positive hal,f-p1ane. Integrals with this type of contour, dencted by c+16

/c-i-

rg"y au ,

'

lemma.

*J-,: ffi'-*".';o,,,

4(-1)'*1 rsin(irr)r[") tr,tr, .,b,.,id1] . ,d,i*1 ...x -n- ) t4.1.3.151 and

O,

and

ux

,=lla" -U-

')-.o lld t-r'. L", ,

(4.2.2)

'

I

hav e

nrl

r t,-r'-1.-(l-uj .rt-l j1.

i#t'

whele Re(a+b+f+g) < I. (1948) page 194. The formuLa See Titchmarsh

' (2nj)'(-1)' -2:) I a,r :nr;.ibni*,,.,*- lli:1-)r1r-r'1r6r-r';'t : t_t "_! lr+r':dr;.;dn;' a

=

[email protected]

d1, .,dr-,;uxr, .,uxr.,) du

t

of this type are

1l.a:*rrrt

*].'i1,*,1r(b-u)du C- [email protected]

'l'his scction on exarnples of special double-1oop integrals is qonclu

O.

lo"

r.rX nJI

'

(4.2.3.1)

lch.4

Contour lntegrals

r{4

-r /.1;3

4.2.4 llelated Integrals

.l.lris section i s devoted to an investigat'ion of anumber of integrals f,unction. of each wi th respect to Parameters of the h ypergeometr-ic those discussed i;a;g;rra and wi,ich are of a different type from involve speaking' strictly not, do allovi. While these results display a they because here included are they integrals, contour j.ntegrals with in dealt the rvlth number of points of si-milarity sectjon. previous thc Wc

-6

-

= - fGl6-tl ":o-?'

(a)

(b')

;

t) L.t +u,c2;. , ca : (d

;

:

I

du

rrc1+"tr]T:O

i . tF- ) (a) zwl-L-L .A:B ; 'C:D, rrc-;T:t) - -' "'- : L,+f-l;c^,.,c-: '''l

:

IL

P.e(cr+f) >

first of al I note the formula given by Titchmarsh (1948) page 187: ? j =--f+---rta;[email protected]

85

Sec.4.2.51

(b');.; (d');.;

-. fnl(b'"') ir, ,,(n). -,'1 (d'-');

,.,r.r] (4

1.

,

.2.4.6)

results expr'es5 integrals of generalised Karnp6 de F6riet functions with respect to their parameters in terms of other generallsed Kamp6 de F6riet functions.

These two

(4'2'4'1) ('

>1, and which corresponds to (4'2'2) in the previous will be the basis of the discussj,on. If the hyperrti, scctions. gcometric function in the integrand of ( (a) ;b, "u,br-u,ba,',bu;x)du I) oF* {4.2.4.2) I(b,+u) l(br-u)

whcre Re(a+b)

i.s expanded in series, then term=b)'-term integration is valid' and (4.2.4.2) maY be written as

I

*lo

(, i ,m) . . (ao'm)

to;,,,l -Gu

*-

du

inner integral is evaluated by (4.2.4.2) takes the form

'i'he

,b r+b

r-2

r(E-lr%rt

b,

oFr(

(a)

means

+b^-1 b, +b^

result c+ i/ cosec c- i-

.,b*;x), (4'2'4'4)

r+,#,br,

provided that Re(b, *br) , 1. SimilarlY, we have a number of integrals involving hypergeometrj-c functions of several variables,

for

example:

-lr-A: B I

ta.)

v

-i'-c,D Lr*r,c2-u,c =

,c r+cr-Z rGr..%rll

-A: B

[-

".,, L1;L,

Re(cr+cr) >

1

(a)

,*,.r,

In

(ar+u)

J

OFU

(ar*u, 1 -a, -u, a3, ., a ki (b) ; z) du

I(.1*r)r(cr-u)

:(b');.;(b(n)) ;*r,

',* ,]

t

,

c+i-

j

{4.2.4 .s)

[email protected]

cosec

Ir(b, .,)

r.|:3;:'

:br+u,br, .,bBi1-br-u,bi,

[t] :

(g)

du =

rrnd

\"

(4.2.s.2)

= i oFr(t/2,1 ,a3,.,ao;(b);2.) Similarly, from {4.2.1. 2) , we have c+ iI cosec In{a- +u) ]OFr(ar+ttaZ,.,aO; (b) ; z)du )-'l

(4.2.s .3) = i oFr(1 ,aZ,.,aA:'b);zi2). These two formulae may be generalised to give lesults where hypergeometric functi-ons of more than one variable are involved' We note the following examPle:-

l_.....-=.-.-.----

,ca: (d');.; (a(t))

(4'2.5.1)

c-16

ldu

'^n_l

very many integrals of Barnes type and related integrals involving special functicns may be obtained frorq the formulae of Sections 4-.2.1 to 4.2.4. A few examples are discussed here. Integrals which are of interest may be deduced by using the well-kno*-n result see Erd6lyi et al. (1953) Vo1. I page 3' If we make use of this formula, we observe that (4.2.2.1) mal' be specialised to give the

so that

of (4.2'4'l),

Cases

f(a)f(1-a) = ?rcosec(ra),

i4.2.4 .3)

ll--l rOP;";tn;;:O

4.2.5 Special

;

(e')

(continued)

.,bi;* ,tr U.zi.s .4)

-i-ii!|[email protected]

.

86

lch.4

Contotrr htegrals

:B-I;B'-r [-ful,r:br,.,bu;b),.,b|r*/zttzf G;c' Lrrl: (s): (g,); -l (con1. ) .

=1F- -A+l .F

(4

.

.2.s;4)

c+

,

(4.2. s.6)

=

[email protected]

I'r(l+c+u)r (-c-u)P.*,r(r)du

*

A number of .j-ntegrals involving simpler hype'rgeometric .functions are nol.r evaluated.. The integral [email protected] (4.2.s.s) t/ (2rl I I(a+u)I(d-u) rF, (a+u;b;x)du c-iis of the form (4.2.L.2'), and, provided that Re(a+d) > O, this integral becomes

[email protected]

c+jm

r

/ .c.;sec[n(1+c+u)lrFr(-c-u,1+c+u;1; c-16

;

r/2

zFt r.t/2;t;[1-z] /2) = The integrals --

2a

Further, if d=b-a and Re(b) > O, L/

we have

g+io ^r1a*r1r(b-a-u)

(2ni) i c- j-

rFr(a+u;b;x)du

the simpler =

I-(:)

form

(4.2.s.7)

ex/2

two integrals involving.the Gau:s functiorl ,F, ar:e investigated. The first is obtained from (4'2'I'I), so that we have a+b+ I IqP2',

Now,

#

i-ri,,*u)r(b-u),F,

which leads to the

#

c-16

'

c+io

andso,

#

'(4.2. s.8)

=

I (a+b) (1-x) (a+b+l) /2

.2'*b

(4.2.s.e)

=

l(a+d) -ra*a Z't

i-.;(a+u) r (d-u) rF, [email protected] c-

equation-s

Re(a+d) > 0.

r'iflbi*la"

c+u.

H-1

'"'

(zi-r/z

(4'

Re(b+bt) 'o i1/(2tr ) I l(a+u)r(b-u)FO(a+u,b-u;d,d';x,y)du, c-iRe(arb) > o and

1/

'

L2)

i/ .r(b+u)r(b'-u)Fr(a,b+u,b'-u;d;x,y)du, (4.2.5.13) [email protected] c+

(4.2'5' I4)

[email protected]

(2ri) J .t (a+u)r(a'-u)rFr(a+u,b;d ;*)ZFt(a'-u,b';dr ;y)du, [email protected]

(4.2.s.rsj

Re(a+a') , O, are special cases'of formulae givert in Section 4-2.2 where Kamp6 de F6riet functions are involved. ltrhen evaluated, the integral s (4.2.5.12) to (4.2.5.14) become, respectively f (b+b'l (4.2.s.t6) --:rfil: ta, D+D ' ; c ; lx+Yl / 2) , L "t'IL 2r-u

and

.

F, ( [a+b1

/2,1a+b+tl/2;d,d' ;x,v)

f (a+al ) a^(a+ar,b,b, ;d,d, ;x/ 2,y / 2) z-

.

(4

.2.s.t7)

(4.2. s. r8l

Za+a' If d=b+b', then (4.2.5.16) takes the simpler forrn =

r (a+d) (L-x/ 2) -b.

(4.2.s.11)

2^*d

'(4-2-5.8) to (4-2'5'11), it is taken that

consider an integral involving a Legendre function. P.*r(z), where the integration is carried out with respect to the parameter

We rrow

[1-z] /2)Cu

c+

t(:H) 2.,"

.

c+i-

In tlre

Z

t/(zri)

it follows that

tr+u)r (d-u) rrr {"*}'b:*)a'., # i-', [email protected]

,

r_---rurA,,

exPression

?..;(a+u) r (b-u) rF, r?ll;irli.lu,

From (4 .2.1.2)

(*

('*u;o-u;*)du =

Special Casec

Sec.4.2,51

r(_P.Pl)

^b+bl Z

1r_{r1-a. 2 -

(4.2.s.}8)

these results may easily be extended .to cover .i-ntegrals of Barnes tlpe where hypergeometric functions of several variables occur in the integrands.

lntegrals bf l,aplace Type

Seo. 5.21

89

5.2 INTEGRAI,S OF LAPLACE TYPE

Clrnp(cr

These

5

of

integrals are, in many cases, readily evaluated by the I of the previous section. Suppose that

Ir- (*)

Infinite lntegrals 5.I INTI(ODUCTION. A THEOREM ON THE INTEGRATION OF SERIES OVER AN

ilrfinjte integrals involving hypergeometric functions, or frrnct:ions reducible to hypergeometric form, such as, for example, Ilosscl functions, were evaluated by the classical authors by means o1' .;rrcli techniques as contour integration or inte::changing the ordcr of integrations. An example of this type of approach is inclicated in the discussion of l{eberts second exponential integlal i n Watson ( 1944) Section 13. 13 . While it may be necessary in certain cascs to employ such methods, the desired results can often be aclticved by the much more easily applicable technique of term-bytcrm integration, provided that this process can be shown to be justified. This neihod has been exploited in previous chaptels of, lhis book where infinlte ranges of integration do not arise. Ih vicw of the theoretical and practical importance of infinite integrals in general and infinj-te hype.rgeometric integrals ln particular, it has been felt desirable that this chapter should be a good deal longer than its fellows Irirst of all, it seems worthwhile to quote a theorem given by Rromwich (193I) page 5OO, to which the reader ls referred for further details. ,4lL"oyCr_1 . 1f the series Ifn(x) converges unifolmly in any fixed interval a: x: b, where b is arbitrary, and if g(x) is continr.rous for aIl finite ranges of x, then jg1*1 1;r,rrx) ldx = aa

Iier.l r,",(x)dx,

provided that either the intesral i lg(*l a

I

(s.1.1)

tIl fn(x) lla* or

lax is convergent. A large number of the integrals under consideration in this chapter may conveniently be evaluated by the application of this generai result' series

17l a

*f.; l .

1

r,.,(x)

i n,.,*",

(s.2. l) con-

[email protected]

dt, r = 'o /e-Pt tr-I I h-tn .t!o n where,

for

convenience, Re(p), Re(a) >

(s.2.2)

O.

It is well-known that i-f a power series such as (5.2.1) converges, it converges uniformly, so that Theorem 7 may be applied and we ha

ve

r = i [r,,,7"-pt t'*n-i dr1. n=U

(s.2.3)

Ct

The

inner integral above nay be evaluated in the

form

f (a+n) p

(s. 2.4)

a+n

of the Euler integral for the ErC6lyi et aI. (1953) Vo1. I page 1.

making use Hence;

r=

(u) p-'

i

gamma

n,., ru , n) p-n

function.

See

(s.2. s)

n=U

This result may easiiy be extended to multiple series, and ifthe coefficient hn is suitably specialised, a large number of results i-nvolving hypergeometric series may be obtained. It is clear that integrals of the type (5.2.2) are closely related to the Laplace transform, and, in fact, the integral I is the Laplace transform of the function t

the

=

n]o t rvhere it is assurred that this series either terrninates or is vergent for all finite values of the variable x. The tlpe of integral to be considered is of the forrn

INI;INITE RANGE

l,lirny

use

Theorem

a-1". n )h t

(s.2.6)

and so these integrals are of j-mportance in applications. Chapter 7.

See

Another general type of Laplace integral is also of importance. This occurs when the function to be i-ntegrated does not possess a series representation with a sufficiently large radius of convergence. Instead, it may happen that the function in question is capable of being represented as an integral such that the order of integration may be interchanged.

lch.

Infinite lntegrals

90

s

mn II r(l-a.-s) fl r(b,+s) 'J ' j=t j=t I ,-t---' z' I(, lzl"t=:-l- 21tL'L ^R,nr-l'j. q p,q'l. P n l(l-b -s) II f (aI nsJ '"j J i=n*t j-m*l'

Vallde Poussin's Theorem, that the equation states whicl'r page (1931) 5O4, sce Bromwich

This last process is the subject of

de 1a

{x,y)dy = I dy I f tx,y)dx abba holcis, provided that both of the integrals !dxl f

/f(x,

y)

dx

and

a

Ji(x,yldy

ts.2.7) (s.2.

B)

b

are convergent, and that either of the repeated i.ntegrals convelges. This,result a,1so holds when applied to contour integrals. 5.2.1 Generalised Hypergeometric Function

qp

.n.r(l*b1-bj), I.f (a.-br) . J=n*l )=l n

I-(l+b--u.) , I l' rbl j=1

L,

of authors, notably !.{acRobert and t'teijer have devised integral fromulae for the generalised hypergeometric function which can thus be given a definite meaning whatever the values of c and D. For further information on this topic, see Erdelyi et al. (1953) VoI. I pages 2O3 and 2o6.The Neijer G-function is defined by the contour integral (5.2.1.2) below. The contour of intcgration L runs from -i- to +i- so that all poles of f(bj-s), of f(1-a1+s), k=1,',n, .i.1,.,il, are on the right, and all poles of integration see Erdelyi paths other For of L. i,.,,'uu thc 1eft, (r1 irl. (1953) Vo1. I page 2O7.

A number

ai'nrrla;', p,q'

lb'

ti

be taken to be a generalised hypergeometric funcThe series tion CFD(xtn;, *h"re k is a positi'"'e integer' When C : D-k, the seriei iepresentation of CFD converges uniformly for all finite values of x, anC if (5. Z. .1) is appl ied, we 1rave the expression

The integral (5.2.1.1) also converges if k > 0 is not an integer, but the result obtained is not so conveniently expressed' If C=D-k+1, the hypergeometric series of the integrand of (5'2'l 'l) IIrs a :'adius of conve::gence of unity, so that (5.2.1.1) does not, in general, hold. In fact, if C > D-k+l, the series in question .loei not converge at al1 apart from the trivial case when x=0. r r \ L^11h^'-,ava* i € n-a n.F the nrtnprrf 6r rrvt! Lollllula lllc lJ, z, t. rJ rrvrur, parameters c; is a negative integer which causes the series under iiscussion td terminate. Otherwise, we have a result vihich, at the best, may be regarded as an asymptotic representation of the integral on the rlght of (5.2.1 . 1) .

(s.2.I.2)

ds.

It seems pertinent to consider separately the cases when m=I and n=1, since the G-function then takes simple forms in tcrms of the generalised hypergeometric function of one variable:

may

a a+k-t.xil, T -sr a-l ,. .k.,- = f(a) J"","'.Fn((c);(d);xr')dt C,rnn{(c},k' k ;ul'S "" S O t5'2'l'1) Rc(s),Re(a) > 0. .l.his result Follows fronr the appl ication of thc gamma integral and the multiplication formula for the gamma function. See Erd6lyi et aI. (I953) \'ol . I Page 4.

9l

Generalised llypergeornetnc Function

Sec. 5.2,1l

=El'

)

[-t *u. -1 '

P'9-i L-*

L',',"r

-b

2,

q, or p=q and lzl O. If' the condition (5.2.3;2) is not met, and if the multiple series conccrncd dces not terminate, then we must employ a multiple contour integral representation of (5.2.3.1) similar to that of, the (i-liunction mentioned in Sectlon 5.2. I . This 'contou r lntegral fornnrla may be uritLen ,..,, I C.l: (d');.; (d(n));

Special Cases

Sec.5.2.41

-

l--

i-

95

io

L/e-st ta- I | /. rnl . jv(tr,.,.n)f (-tr). .f(-tn) o . kxr.) L-t- -1ox 1-r -t, 1..(_r k xn) r-1 "n dt

I

.

.

dtn_l dt

,

(s.2. 3.8)

i

I

r.,. ro,.\. "i',11'\'./'\ts r,''(t r.(n).,.*r,',*n ), _l .

i

Re(a), Re(s) >

If the order of integrations is interchanged, wemaythenevaluate the inner gamma integral in the usual way., andwehave theresult io

X = Ls-a

rllr"r ii,.,,.,.n)r(-tr)..r(-rn) -1€

(-x - n-)tn dt-..dt I n' (s.2.3. s)

[-xr)tr

[email protected]

FG

, r(f;) n [f(c])

. rer(n) I I

l=1 ' t=1 CD , r(..1 n tr(dl)..rta!n))l(:ni)n )' J '' j=I i.j=i' CD

(s.1.3.6)

(.j*, l*.*tn) .n, Ir(dj+tt) .r(a!n)*t ll ji-j=I .

and

V(tr,.,tr,)

=

=lf l=I h(,

n,

.ll(tr+tr*.*tn) )=t l=r .

(nl

io

/. frl./v(tt,.,tn)r(a+kt,r.*htn)r(-tt)..r(-tn)

-jo

-i6 -Y"t.t.

. -X..n-t n dtl..dtn. * (, l..t (5.2.5.9) kJ kJ SS This last integral is a specjal case of an n-fold generalisation of the lleijer G-function. 5.2.4 Special

=

O.

Cases

vast majority of the special functions of appiied mathematics be expressed in one way or another as hypergeometric functions and only a few representative examples of Laplace integrals cf functions of this type will be disctissed here. The

may

inregral (5. 2 . l. 1) in which the integrand is taken to involve a confluent hypergeometric function. l{e thus have . -st a-l -A J. -' t* ',F, (c;d;xt)dt = f(a)r ' 2F1 (a,c;d;x/s) (5.2.4.1) 0,., by the use of (5. 2. I . 2) ; Re (a) , Re (s) - O. Consider ths

of further special cases of this result may be deduced. that x=s, then the Gauss furrction onthe rightof (5.2.4.I) may be summed by Gauss's summation theorem glving the formula ? -=t t a-I - (c;d;st)dt - r-(a)r( o. F

inner i-ntegral

rnay

be evaluated as a

gamma

function, and so

we have

I

=s -"(r)

\-l

The Y--l

r

fal

1-

16

I a'. S [email protected] -1o

r (c+u+v) r

(et"ryll(s1lI{! '+'.')r(-u)I(-v) I ( f+u+v

)

, i-x/s)u (-yl=)v du

dv

{s.2.4.28)

In order to obtaln a representation of this last result in terms of convergent series, the above integral uay be written as an integral of Barnes type of a G-function of one variable. Some rather lengthy manipulation eventually leads to the sum of six double hypergeometric ser.ies of higher order with arguments s/x and s/y.

lnfinite Integrals

100

lch.

s

If tlic function Ft under consideration terminates, then the siturtion is much mor6 straightforward, and only one terminating Kanrpd de F6riet functj-on results; /c-st ra-' Fr[-N,d,d' ;f;xt,yt)dt 0

(drm) (di,n)Imyn ."***n-1 dt = y(-Nr+1n) L m:nl i"-tt (t,m+n.)

(5.2.4.2g)

'0

r(a)

=

s

(s.2.4.30)

a

Re(s), P.e(a) > O. Whcn special cases of the generalised Kamp6 de Fdriet functlon.of scveral variables are considered, we obtain the following results which give integrals of functions which are convergent for all finite values of their variables:-

i"-tt .'-' .jnl (bi,.,br;c;xrt,.,xnt)dt

If it is desir.ed to evaluate a Laplace integral of a non-terminating Lauricella function, the si-tuation is muchmore complicated and vre must tackle the problem in a way similar to that.outined in deallng with the function F. in the expression (5.2.4.27). Of considerable practical importance are the Laplace j-ntegrals of certain cases of the generalised Kamp6 de F6riet function whilh consist of products of several single hypergeometric functions. For examole. r_ :br; ';brr; T -st t a- I ho,rlx,t,.,x Jo. -O:1

Je 0L

I

a-1 - (hr -. ;d, ;xrt) rFt

=j. -st 0

=

II*1"

(5.2.4.3r)

r -sL a-l J" -- t' 'oFr (-idti*r.-)..cF,

(s.2.4.32)

0

(s.2.4.33)

5

Re(a), Re(s) > O. As in the case of the Appell functions, when the Laplace integrals of terminating Lauricella functions are discussed, we have such formulae as

I{c(n), Ilc(s) > 0; see (5.2.3.1). -- f*

3s)

(-;dn;x't ,21d,

hypergeometric functj-ons, such

(s.2.1.36)

'

z

as

= ze ._t1

'.

F.

(i;2:22)

{s.2.4.37)

,

r -' t-a-l' sinh(x,t)..sinh(x tl dt [e-st n' 0 2n x-r ..x nl(a+n)'-[nJ i2'2'''2i ;---i-n-Fi"' {'*n'I'I'''l ts+x1+ ' +xn] +. +xn], +xrrJ , ..,2xr/ 2xr/ [s+xr+. ,2: [s+xr+.*xr-rl ),. where

Re

(s+xr+ . *xn)

, Re

rIe -' t- ' Jc-'I(x,tl..J c (xnt)dt n 0.1

(s.2.4.34)

(s.2.4.38)

(a+n) > 0.

Siml 1ar1y, .

au-t aln' (-N,b;cr,. "cn;xrt,.,xnt)dt :c1;.;cr.,; I' ' 'n'

(s.2.4.

we have

i"-tt .u-t o {") rur, . ,b,r-1, -;c;xlt,.,xnt)

a 'O:I'

;xnt)dt

In the expressions (5.2.4.35) and (5.2.4.36), Re(a), Re(sJ > 0. Since the hyperbolic functions may be expressed as confluent sinh

i"-tt '0"

(br.,;d.

@/2,la+r)/2;dr. dn;4xrls,.,Jrn/s)

tP[.)

and

= L? rji) r",rr,.,bn-1,- ;c;xr/ t, . ,xrr/s) ,

rF,

0

=

/"-st .a-' *in' (b;dr,.,dn;.x,t,.,x n rldt o' .,d,,,;xrls, . ,xn/s)

..

"J

(a,b1,. ,bnid1, . ,dn;xrls, . ,x,,ls)

S

= ,19 r[n) cu,b;dr,

I

:dt ; . ;dn;

and

0

r(a) -(nl . Fi"/ (arbr,.,bn;c;xrls,.,x = r? n

l0l

Special Cases

Sec.5.2.41

-.f

.t[n)

a

(xr/2)ci

1

. . {xn/2)

cnf (a*cr*

=

f (cr+1) . . f

(cr.,+1) sa+c1+'

.

+cr.,)

+cn

)a))

f Ia*cr+ .+cn1 /2, [a+cr+.+c,.,+1]

,.,-x-/s ), /2;7+cr,.,I+c ;-x,/s I It

Re(a+cr+.*.r), Re(s) > O.

(s .2 .4 .3e)

Intcgralc [Ch. 5 polynomial of the first kind may be expressed Inltnltc

102

Now, tho ChobYshev

qs s Gsuss function by moans of, the formula 1-+

rr(z) = rFr(-n,m;l/2;-)

Thus,

(5

(5.2'4.4o)

,2.3,4) gives the formula

6

Jfe'tit ta-l T*, (l-2xrt) . .Tmn(1-2xr.,t)dt

.

r

rOl- nl,? i;:-mrml;'i-mn'*r;*,rr, 'O,rL_: L

r/2 ;.;\/z i

,,

rra) (d-a,N) r -St t a-1 (5.3.3) je = ,Fr(-N;d;st)dt : (d t't) su , 0 Bessel function of a modified involving An interesting integral the second kind 1.' r.i.) dt = 2a-L r(1-i5r(=1), Re(a1c) > -1, (s.3,4) 0

,.rrJ

,

(s.2.4.4r)

Ite(s), Re(a) > O. 3.3 INFINITE INTEGRATS ASSOCIATED }VITH CONFLUENT HYPERGEOMETRIC FUNCTIONS AND BESSEL FUNCTIONS

has been given by Luke. See Abramowitz and Stegun (1965) page 486' This may be obtained by replacing the function Kc(!) byits Barnes in1egrai j-n the form of a G-function, and reversing the order of. intelration. The resuits (5.3.1) to (5.3.4) will be used in what fo1 lows 5.3.

I

.

Generalised Hypergeometric Functlon

The integrals to be consldorod now aro mo$tly of tha form

class of infinite integrals includes those whose integiands involve a ionfluent hypergeornetric function along with anotter hypcrgeometric functior. Sin." the exponential function and the Bessel .functj-on are particular cases of the confluent hypergeometric furlction, Laplace transforms, Hankel transforms ani Fourier transforms may be included under this heading. Certain devices for the evaluation of suih integr:ats are. givetr by Watson

iu-t' ,'-',F,

the hypergeometric functi-on(s) of the integrand of their argurirents and integrating term-by-term. the hypergeometric function(s) by their Barnes integral representations and reversing the order of lntegrat i on . Watson (1944) Chapter 13 is a frequently-used source of information on irrfinite integrals involving Bessel functions, and such integrals arc of inpor*-allcc in nany branches of applied mathematics as well as being of interest to pure mathematicians,

in the

Art inrportant

(1944) page 381, (i) Lxpanding in powers (ii) Replacing

r03

Generallsed Hypergeornetric Function

Scc.5'.3.1

and these include

(c;d;xt),,1.,((f) r (g) ;vtk)dt, tts(n),Ro(n)

r 0, ($,3,1,l)

If l: g (i+k, wherc k ls c noritlv$ lntol|$t', tht gonerul llertl lrypttr" gcome[ric funct ion of the integrantl convsrgsn un,l f?ormly I'or u I I ln iinito vElucs of thc variable. If this functLon is oxpuRdcd of series, wQ nay then integrete term-by-term. The application (5.3.1) cnables Lls to evaluate the inner integral which occurs .

form

f(a+Ltn)

(5;3.1.2)

rFr(c,a*km;d;x/s) provided that lxi . l.T. Hence, (5.5'I.1) may be t+ritten as (s.3.I.3) (xs')^^. l1ely' -,,.--a,'(yr-o)"' " ,u " ( (g) ,m.J mlnl If k=l and F < G+1, a Karnpd de Fdriet function results:

The formula

r -St a-1 (s.3.r) Je t ,Fr(c;C;xt)dt = I{P 2Ft(u,c;d;x/s), 0 Re (a) , Re(s) > O, has already been gi.vcn, see (5.2,4.1),. and its special case (s.3.2) (c;d;st)dt t a-l ,F, = L(3)l'(d)r(d-c-a) , r' r se r1a-.)r(d-a) 0 Re(a), Re(s) > O, Re(d) > Re(c+a), see (5.2.4.2) If c = -N, where N is a non-negalive integer, then the third ccndition of convergence'of (5.3.2) may be dispensed with since the series involved terminate, arrd we ha.ve r -St

Je

(s. 3. r .4)

If c = -Nf the condition F s G+l rnay be relaxed. Suppose; further, that x=s, when the .inner iniegral'mentioned above may be given in closed form by means of (5.3.3) . The integral (5 ' 3 ' I ' 1) then --l becomes a+ft-1 1+a-d l+a-d+k-1. ,. l-- ^. 't"yl rfa)fd-a.N) l(rj'k' ^ J' I

F 1aj-

F+2khG*k

Lrrl,riiu,,. .,]:e+Y=! ; .k

(5'3'I's)

104

Infinite

Integrnls

[Ch.

interesting special cases of (5.3,1.5) Section 5.3.4 for a few examPles.

Man)r

5

may be deduced, see

A formula similar to (5.3.1.3) may be deducec for the evaluation

of the

i nte8,ra

1

(s. 3. 1 .6)

..r.[*t)FFG((f); ig;;ytk;at, "-tt .^-l 0

f

by employing the formula (5.2.4.12) ' The expression (5.5.1.6) bc written in the form

lGiQ-G4): -aTE(c*1;

tTizll (.(Il, l,) ({/ sl) * ( ; x2l tas ; ra*c, ((g),m)(c+1,n) m! n! '

2

I)

n

may

, (5.3.r.7)

F-k-l : G, un) ess the function pF5 i-s terminating. In addition, Re(s), Re(a+c) > O. Furthermore, if we let k=2, the result (5.3.f.7) nay be written in the form of the Kamp6 de F6riet funct ion r(a+c) (x/2)c

f(c*l) sa*c

Fi,l;g u:u;l

H

rl

l+,o?'; :[gj;e+r' [;] ;.;,,,r, il ttL

"'xt,yt)dt (c) ; (c') ;x Il (d-a,N)r(a).B+2:C;C'.(b),a,I+a-d: \ J F+1 : G;c' (f ) , l ta-d-N: (e) ; (e') ; s's' ''

Re(a), Re(s) > o and B+C < F+G and B+C' j- F+G', unless the de F€riet function on the left of (5.3.2.3) terminates.

.4.I2) and (5 .3.4) nay be used respectively to obtain the tir'o results now given. (d')i.yt2,rt21dt T -rt t a-t -B:D;D'r(b): FF,c;c't(rl (d); The formulae (5.2

0

=:

i.'

where Re(a+c) > -I and k is a positive integer. If F+k 1 G, termby-term integration using the formula (5'3.4) gives the following expression after a 1itt1e reduction:-

A=

2a-ttfg#llrCgjal

f-,., a+c+) a+c+k a-c+l

f[a+c)(x/2\' /-

1.(.-,) f-.c

a-c+k. ;

6

L

'

-'

=

2a-trr,*!*1rrre:*llFB:2::::l r 'u'u [(o),"*'"*:

'

0

(s.3.1.r0) \

;+*,+y

:(g);(g');

dt, (5.3.2.1) I m,n=u^nr.nt'*'

of the previous section may be generalised fairly to apply to integrals involving hypergeometric so as easily functions of several variables. We thus obtain the formula

r (a) (d-a, r") -B+2 : C =---_r.'F+l:G ,u (d,N)

Re(s), Re(a) > O,

[:?],i":]; :::;:l],.,.,,,.,,.]

o,

,a,I+a-d: (c');., (.(n)),*t ,.;l [::t ,t+a-d-N:(g,);. ;(g(n));s "f ' (5'5'3'1) B+C

I

-l

(S.j.2.S)

5.3.3 Multiple Hypergeometric Function

form

whcrc thc doubte series of the integrand is either terminating or rrrri fornrly convergent for all finite values of the variable of integrrr1 ion t, the term-by*term integration is justi.fiedand we have

\-l

(d) ; (d' )

I

i"-=t .'-t,F,(-N;d;st)rf':

o-1 u = Je -t tu-l,rr{-x;d;rt)

[rrl

Re(atc) > -1.

5.3.3 Multiple Hypergeometric Function

--

I

R'n'nr .(b): (d); (C'); "t2.rt21dt {.- *.(t) F;;;;;,(t;j; i;j I ir,j'xt-,rt

The methods

@@ r -Sl -

I

T,

If the pF5 function of the preceding integrand is terminating, then the iestriction F+k < G rnay be lifted. If the integral under consideration takes the

::(f):-:-:

2 t= . 2 x-.2, - ;4v' , ---=,-!'a (c);(g');c'I; ,2' ,l' .2-l (5-3.2.4) . .

and

1

'zk ''' 2k _ou*1. - lut'-zv "'2x (e) * n*zttc ^ ^J ' l_

, (ej ;(g'), [-"-. r*e+i l],]r::(b):-:-:(d);(d'); -(3) F' I I' I )--

'.(xtJ

Re(s), Re(a+c) > C

K.(.)nFc((f);(e);*t2k1d., (s.3.i.e)

(s.s.2.3)

?Jr)

where

This section is concluded by considering the integral

o=

.3.2.2)

(s

A;;u-l"''n

j"-,. .,-t. F, (-N;d; st; rf ,E;E i, [?] ; [;] ; [;: ] '0

1"

(s.3. 1 .8)

(I+a-b'm+n) _ (d-a,N)l(a)Ih- _ E4;l) -lT*a-E-N'm*n;'

by the application of (5.3.3). It is now assumed that the dout le series is a Kamp6 de Fdriet function.This nowgives the expres s ion

Kamp6

wl'rere

105

Double tlypergeometric Function

Sec.5.3.21

s F+G.

[(lh,

Infinite Inte$als

r06

5

of (5.3,2.4) to multiple series does not yield an elegcnt exPres sion, but before proceeding to the discussion of special cases of integrals associated wi!h Bessel functions or confluent hype rgeometric functions, we note the following result which is an ex tension of (5. 3.2.5) ;-

The extensirrn

o

{.o

*"

(t)

FF:3

ftu1,1a,);.;(d!'1r,,. t2...x., ' n

L(r),(e');.;(s(nl),'

I o,

(n). . . :. ,, '' tu . )'4x- . ',0*J , * 2 .' 2 :(g');-;(g(n));'"1' [tt]; ) ";',3 g+[+l (s.s.3.2) Re(atc) > -1, 5 F+C.

A number of special cases of (5.3.i.4) which are of some lnterest First of aL 1 , if r're let F=G=k= I , an integral t-rf a pair of confluent hypergeometric functions may be evaluated:

may be deduced. 6-

j"-tt .u-', F, (c;d;xr) rF, (f;g;vt)6; = tGJ-e rG,c,r;a,g;l,rJ, oras

(5-3.4.1) e(a), Re(s), O. When the parameters and variables are further specialised, as appropriate, we have several more results. A few examples are now given

rFr

(f;a;yt)dt

(-f-)t ,,l-(.,a-f ,f ;d ix/s,y/ [s-y]). - r(a) a 'y-s-

(s.3.4.2)

5

next consider an integral involving the product of a pair of Laguerre polynomials. This is

We 6-

/"-tt r^-t

lfitst)r$tst)at ,\ rr

I (a) (c+1,N) (d+I,M)

lft#*i*q rFr(a*m,-N{;d+1;1)' (s'3'4'3) It! ,' N! The inner Gauss function may be summed by Vandermonders theoren ISlater (1966) Appendix III], so.that the right-hand member of (5.3.4.3) becomes =

F Il0-(ctL-Ul,Gr,Le-,Iil - 3'-2 *fl Nl Ml

(d*l

",M)

be summed by

,

If F=0, G=l and k=2, then the oFa function in the integrand of (5.r.1.1) may be replaced by a Bessel function. Thus

?-.. .-'rFr(c;d;xt)Jr(rt)dt a-I-. i"--' 0-

ts. 3.4.4)

UUtltrA-

(s. 3.4.6) H,q'(a+B,c;s+l,d ,-{,r., , 4s' t ,u*8r(g*I) Re(s), Re(a+g) > O. H, is a Horn Tunction, and jts scries representation js given by Erd6lyi et a1 . (1953) VoI. I page 22'> as mn (a,2m.nJ (b,Ir.)- x J_ (s .3.4 .7) Hq (a,b;c,d;x,y) = ) (c,n) (d,n) nln!

=

5.3,4 Special cases

,0

="(c-M) ( -M,N) (c-l-2M,N)

107

Re(s) > O, Re(c) : Il-N.

a+c+1 a-c+l .;[u'J ., ,.

? -.t 1'-' a-l - - ;d;xt) /c-" rFr(c

Ce*r

If a=c+N-M, the preceding Clausen function may Saalsehtltzts theorem and so we have T -.t-' t"c+N-M-l -c. L[ (st)--d. L[(st) dt I"

-l

.2a-L r(g2r5r(e+!-)

0

$poclul

S0c.5.3.41

of an integral involving a l;aguerre polynomial and a js now given. @ T .lt, .'-r r-l(st)r*(r -2yt)dt - r(a) (c+1:a'n)rt, f"-tt L*'*'u I . --lla-c-n.'I nl sO L7' ?.-' r (s.3.4.8) tt we consirier (5.3.i.5) with F=2 and G=k=i such that the Gauss function in the integrand is tdrminating, an integral involving a Bessel function and a Gegenbauer function (for example) may be An example

Chebyshev polynomial

investigated.

i"-tt .u-iJc'(xt)cl1r-zyt;dt - rr 6 I (rd,") t(rf.) (l O. A function related t9 H4, but of higher order, appears. If

d=-lt2-n, this result simplifies as foliows:Fe-st ta-lJ. t*t) cr.,r/2-nti -2yt)dt

(s.3.4.101 (cont inued)

[Ch.5

Infinite lntegrals

I0H

(corrtinucd)

4tr.

(s.i.4. ro) -t -2n.n'1 l(a+c) (x/ 2)' lf ".:,ulz;D!::l-udr(c+I,PJ m:P: 1 Asz ,o*. n ! f(c+1) 'l'lrc series on the ri ght of (5,3.4.10) terminates in y and conv(

for

crges

l*

I'

z l=

l.

A further example may be furnished by the integral of a Bessel Ii-rnction and a Legendre po1ynom.ial, .u-

i"-tt

1.1.

0

(s.3.4.1i)

(xt) v nlt -zy2 t2) dt.

'l'his nay be evaluated in the form

- i,.- xl21 +,u;L,-n,l+n; , i,r..rr.-rl '0,I ;o l,' -' -. I ;1*.;' ' sRe(a+c), Re(s) > O.

r(a+c)

(x/2)c

F2-,2.;91

109

Special Cases

Sec. s.3.41

Integral s involving confluent hypergeometric functions and the confluent Appell functions may be evaluated as speclal cases of (s. 3. 2. 3) . For example, r -st t a-l I (-N;b ;st)v Je 2(c ;d,d ;xt,yt)dt ,F, 0

(b-a,N)I(a) -3:O;O,c,a,1+a-b:-;- ;..

..

ffi,*i,;;i(;;;:u-l.r".a,a',*'y),Re(s)'Re(a),?,.,.4.|7) If c=l+a-b-N, then this expression simplifies so that the rightfunction of the fourth kind. This is hand member now involr'es an Appe11

(s.3.4.18)

15.5'4.Lt)

|

F6riet f,,rr-r.tion above con\rerges lf lxl t itl' llcforc procccding to rhe discussjon of special cases of integrals of BessLl functions and double hypergeometric functions, wc investigate integr.als invoLr,lng a modified Bessel function of the ,o"oid tind and firsrly, a Eessel function of tl.re first kind, and sccondly, a Jacobi PolYnonial

'l'he Kamp6 de

i.'

*.(t)Jo(xt)..

=

fi#+{.u*o

*"(t)oFr(-;d+1

,.#dt'

(s'3'4'13) (b-d,N)r(d)

.fhe integral on the right of (5.3.1.13) may be evaluated by of (5.:.f.lO) and we obtain tlre well-known result

[t.,, K rt)J.(x-)dt ' o' '

0-

means

i\- 2 )r'1.ra+d-c+1. ='L,^-lr1a+d+c+.l.., z )

*

2F

L,-

,q-!,uagfaf;d*

r;

-x2)

.

(s.3.4.i4)

ro (t)P'"(l-2xt)dt, n

(s. 3.4. 1s)

0

ing that the associated Gauss function is terminating' It thus l'ollows that the above integral nay be written as 1.,, -c+ I .,ir , I ,.r(,.lr+c+ 2 )',,...a Z , x4 I; ( -n, f+1+g+n, Ia+c+1 ) / 2, la-c+1) / 2 ;f*7; x), Re (atc) > I

rrot

il.

r.

(s .3 .4 .20)

RelsJ, Re(d) > O.

Furthermore, the Kamp6 de Fdriet function on the left of (5,3'2.3) be so special i-sed as to give i;lcgral s invoJ.ving pr:oducts of three confluent hypergeometric functions such as

/e-st ta-1 r Fr (-N;b ; st) 1F1 (c;d;xt),

'l'hc ,lacobi polynomial. below may be written in hypergeometric form irncl so, (5.3.i.16) may be used to evaluate the integral ct'

=d

F, (1+d-b,c,C I i 1+d-b-N;x,y),

rnal'

Sec Watson (1944) Page 410.

:" /It'K

(b,N)

a. O. rUl

a,-

F,

(c' ;d' ;yt)dt

_ (b-a,N)l(a) (b,N) ,' The integral (5.3.2.3) nay also be specialised. ln several other ways, most of whlch yield rather cumbersome forms of the general triple hypergeometric function of Srivastava. However, if we consider the integral of three Bessel functions of the first kind,an elegent result fol1ows. This expression has already been given, see (5.2.5.39) when n=3.

lch.

lnfinite lntegrals

I t0

=

*fy9 2u-l

ttgr,r.gr -,a-f+g-c+1. -.a+f+R+c+1,

2

r(f+1)r(g+1) )

Z ;-v.*, -y-), Re(a+f+gtc) > -1.

...s+f+p+g+la+f+q-c+l *,'o ( ,Ttf+l,g+I ^,

2

','',

)

(s .3 .4 .22)

SirniIarIy, we may evaluate the following integral involving the I)roduct of a pair of Hermite polynomials:-

ft'

x.{t)Her',(xt)Her^(yt)dt

= qP'l?#(-2)-'-*

0

,^,\ rra*c*Ir.ra-c+I.r12: I ;I l-9:S:l," 2-),t 2),0:1;tL._.

2^-r

g.i :-n;-m;

,lr]r, 2x,2y l

a-1 /c-'" t"tFt

l--=t l.

-u-l .

,'

where

(t)J,q-

(b,N)

(s.3.4.2s1

we have an

t) . .J d.

-l

*.

+dr.,+c+1. a+d,

(s .3 .4 .26)

cr, .,cn; 1+a-b-N;xrls, .,x,.,/s), (s. 3 . 4.27)

see Lauricella (1893).

rl Ll -.i I n

I

(s .3.4 .2s) 5.4 II..IFINITE INTEGRALS AS INVERSES OF BARNES INTEGRATS

A convenient and practical wa1' cf deduci-ng certain types of infinite integrals involving hypergeometric functions is to make use of the l'le11in transform thebrem, see Siater (1966) page 148:

.

S-'l

Jx"^f(x)dx,

(s.4.1)

(s.4.

2)

0

0,-

(b, N)

'-

+.+d -c+1 n

then g(s)=

integrat involvjng the function 6.:-

I (e) (b-a'N) n{n) ' l) 1t*r-u,

r

-t. c+l.-/ x - g(s)ds, rf f(x) - lTl' ^ . C:1-

*"t first defj-ned by Lauricella (i893).

the Function f[')

(x,

n I^a-I x_I..x 1n (x t)rrt - lfd-+I)..1[d +I] .' n ' '-t"n

(s.3.4.24)

f"-tt .'-1, r, [-n;b; st) ojn) t.r, .,.nia;xl t, .,xnt)dt ,'

L

L i..rru1r . run f, ... + - . ,,-L.^.\,,,(n),,.^ ur 5tJ Y2 ( 1id-u-.\ rA1u r . :^ n 1, 1\-.\;

SimiJarly,

-

x-

O.

f (a) (b-a,N) .(n), r^L 1a,1+a-b;dr,.,dnixr/s,.,xr",/s), =- . '

.3.^.28)

I

0

Re(a),Re(s) > It thus follows that

lne '-f.u

" .ln) L---- z

;. "'a,,.,d,ix,/s' Fi;;(i;;:;-* "3:0rc,a,)+a-b: "x,,/s) '

(s

This section is concluded by giving an integral involving several Bessel functions of the fi.rst kind and a modified Bessel function of the second kind. This integral may be evaluated by specialisthe formula (s.5.3.3). dd

(s. 5.4. 23)

which may be tackled by using (5.3.3.1), when it will be seen to bc equal to

Fffi

or

l-"*o,

fnr (c;dI,.,dnixtt,.,xnt)dt, (-Nrb;st)v)"'j

l(a) (b-a,N)

:x t)dt /e-st .a-lrFr{=tt;b;st)rFr(.r ;dtixtt) . . rFr(c n':d n -n' (b-a,N).2:r I a'l+a-b:cI;' ;cn;x,/s, "x ,/'] . = r(a) f G") 'I:l l-1*a-b-N:dr;';dni

0ln

Re(atc) > -l: Wc now consider the integral . -sf

lll

as Inverses

In addition, we have an integral involvi.ng the product of several confluent hypergeometric functions:-

irrtersting inregral involving the product of two Bessel funclions of thc first kind and a modified Bessel function of the sct:oncl kind may be evaluated as a special case of (5.5.2.5):-

Arr

It" r (t).1.(xt)JE-'(yt)dt 6 c'' t'

Infinite Integals

Sec. 5.41

s

provided that g(s) exists in the Lebesque sense ove-r the range iero to infinity, By a simple exponentlal change of the variable x in both integrals, the Laplace and Fouri.er transforms may be deduced.

is a Barnes contour integral, and many cases in the literature. Probably the most important general class of integrals of this type is the I'leijer G-fuiction. This function has been discussed in Section 5..2.1, and its defini.tion in terms of an integral of Barnes type is given as equation (5.2.1.2) .

The integral (5.4.I) have been discussed

lch.

Infinite Integrals

l12 ltJe now

quote an important theorem

Slater ' s

t(z)

=

l-?L:red

#-,

Integral

T!teo:911

I

fo Slater (1966) Page

due

s

143.

f

(cont inued)

u

*

|

(a) *tr

v,

(b) -hv, 1g) +hr, (h)

"l,t L('i.n v,(d)-hv,(j)+hv,(k)

r[:]::: [:] -:: [3]l:; [] -:, (s.4.3)

113

lnfinite Integrals as Iriverses

Sec.5.4l

V\

'

-h,l

-hl VJ

((a)+hr,n+n) ((b)-hr,m-n) ((g)"hr,n) ((e),mJ +hr,m+n) ( (d) -hr,m-n) ( (f),m) ^fi+Iv -TfrflnI((c) ' mrn=U' (H-x) (1*hv- (k),r1**rhr*' (-i)"

i

,

" where

the series A+B*EF.+D+r(x) is absolutely and uniformly TI

+(A+G+B+H-G-D-J-K)

*I

(111U_ (.1 (b)*ar,2m*n) ( (h)*.r,*.nl f(") :l) :i) ( (d)*au,2m+n) ( (f),*) G.X-G)rill'*u- (s) ,m"n)

(

m, n=O

G

+)I r

u=I

rl -rr,'(b) *su, (e) '-e, l-f

(i)

l_f

either (a) (b) or

m!n! , (h)

, larg

con-

z]

we have

t1+a -u -(i)

((k)*ar,*',,)

(5'J'5)

I

and (ii)

-c,

I(z)'= I(r)' A,*

rvhen A+G+D+K when A-+G+D+K

> B+H+C+J = B+H+C+J

L(z)= Il,)' Ro

I

.l -er, (d) *su , ( j ) -er, (t ) *er,l

either (a) (b) or

when A+G+D+K when A+G+D+K

(s.1.6)

If.l

B,-

anO.

This theorem has been established in detail by Slater (19661 page 143 by the use of contour integration. A11 the results of ihis section may be extended to multiple integrals and multiple series by appealing to de la Va116e Poussinrs theorem; see (s.2.7) and (5.2.8).

$.4.

lch.

lnfinite Integrals

n4 I

s

i*s-t.m,"r*l'j.,0* p'q ' ,0, '0

=

mn l(bi+s) .ll, I(i -a. -s) .-j=r j=\

'

qp

where SO

-5 Jx- ^ oF.((a) ; (c);kx)dx =k 0

_r-I

_l-r

AC nr(ai) TTI(b.-s) j=I ' j=) )

and

0

where

D

(s.4.1.1) 5.4.2 Double rlypergeometric Function

Within its region of convergence, the Kamp6 de F6riet function of trqo variables ma)- be expanded as a series of generalised hypergeometric functions of one variabl e:

(s.4.\.2)

(s.4.1.3)

r m=0

G=B=U=J=D=K=C=0

' [i]

e-<

fc'\ .

-q r-'

when we have

ds = z-a

Re(a-s) >

(s.4.

s)

, larg zl

1 Similarly, if we 1et G=1 and A=B=C=D=E=F=x-0 and G=l and Ii-C=K=F=0, it follows respectively that

\

1.

r,;ri, r [:] :: : [: ] ::, -' :

J

(s .4 .2 .2)

(a)*s, (b) r's*sF6*p ( (c) *s, (d) ix)ds

-io

.(a) : (b) ; (b') ;-. -A:B:B' (i;j j,*',),

t,;;;,

;

(s.4.2.3)

t;j ; i;,

1 C+D+Dr+1 and |ln-n'+l-C-t)tl , larg fl On inversion, we have

.

:*,rrar = .t[:]-i: I,'-'ot;B;3:,[l] [:]; [:l]

(c) , (d')

:

A=H=J=0

(s.4.2.t)

:rt

where A+B+B,

result

= r(a-s)E.rFF(u-ii,t"' i,.,, 'rI

O,

: [fl : ]

x.*n()r'r([;]; - -) z'

-io

i,'-'-1",p()rrr([;] ;-).,

r

(s.4.1.4)

wl'rich on inversicn gives the

I

_ [email protected]

consider more general types or MeIlin integrals which are consequences of Slater's l{ixed Integral Theorem. Take A=1 and

Wc now

Z.;fJ r(a-s)r*tFF,- i;f "'l*r

(ar,,,fiG[ffir

If the inner hypergeometric function is replaced b1'' its Barnes integral, the processes of integl'ation and sumnati.on may be interchanged, so that the right-hand rnember of (5.4.2.1 ) may be written -m +s,-slrt dr, ( (a) ,m) ( (b) ,m)X , (c) *m, (d ') ., i 'f :,. (a) 'm*s , (b' ) ((c),rn) ((d),m)m! "' t[")*,n, (b') I 2njJ', (c)+m*s, (d') +S Hence,

i--

-m

[ (a] ,m.) ( (b) ,ml x

*!6

in this theorem

zl

r s-A-i

where O < Re(s) < nax Re(ar).

,

> H+J-1, Re(g-s), Re((h)+sl > o ana |(I*H-J-K),larg

K

1al an l. j'* llo note that the sPecial case of the G-function, (5.2.1.3) , after a strai-ghtfonrard change of variable, enables us to write

i.-r

(s.4. 1 .6)

JL

whcre, for convergence,

CA [r(ar-s) r(s) ITr(c;) J ;-r ) r

115

Double Hypergeometric Function

*at (h) +s, (i ) -e, !k) r'. - .rL,B-s, (j)-s,(k)+s,(h)+g

I f (I-b -s) Ilf(a +s) j=n*l j=m+l '

Re(s) < 1-nax Re(a.)

5.4.21

(t Irs-s- r.. -' -r) r-t)d, -r...r1.s-!ii:!l')*eir-rl (k)*s Il+J K' ;' L-

Gcneralised rlypergeometric Function

Iu ordcr to clcduce l\'lel I in integrals of the type (5 ' 4 ' 2) from Slatcrts lrlixeH+C+J, Re(a-s),Re((h)+s)>0 and I lr*H-c-.1-xl,larg zJ .

j,.-t-,

(e) ; (h)+g,1+g-(j) oB:E;H+Jr(b)+g: \(d)*g:(f); (k)*g

'n:F; K

a-

;*, Gt)l-J /r)a,

tti_i;

l-ru)

**,* .*rr,- l*.,

where D+K+l>B+H+F, Re((b)+s),Re(g-s) ,Re((h)+s)

larg zl.

A.D Fh''

x

'C:D

,

(t) *., ;

(a(n))*u

'**., l*''

/x ,n

ied

5_1

and

B -A: t'c,D

A +B

.w e

[,,, (u

c)

-s, (a

B ,A: 'C:D

--^..;.r^, Pr vv rusu

(n) (n)

;

-< C+D+1 una

>

I a''

I

*tdr.

ln

(s.4.3.4)

_]

l^rs x

I

^t

maY nOr.' be

]ra ve

:(h,)r.,iu(n)), I :

(d' ) ; . ; (a(n))''"''

)

-s, s, f.l , to t')Jl

"'n-l

o*n

)-s, (a), (o(')! (b');.;

(btn-il,

I, t') 1,., -s:(d');. ; (a(n- , ''''*n-r-i l-r

.'l

for-rn

:'xt'-l (d,') ;. ; (a("-r),'x1' L. c)+s:

Lr.t

0

-

(s.4.3.3)

ilA+B+t-C-Dl The l.Ie 11 in tr an sform theorem, see (5.4.1) and (s.4.2) ,

5.4.3 li{ultiple Hypergeometric Function

First of all, we give a single Barnes integral for the generaliseo Kamp6 de F6riet function of several variables. The function A:![(a),(b');.'!o:"]);*. (s.4.3. r ) %;;L;.;,i. , iar,r, ,. ,xl rl may be expanded, within its region of convergence, into the (n-1) - fold series of generalised hypergeometric functions of one variable which follows:-

*

1b

, i- [r,t*=,1b(n),.., =l .,

provided that

r t( ='1,

>O

(5.4. 5. r) in the

thus rtite

' 1,, ,rr,r,.rl -il,:L(.)*,, (a(n))*, lr a)+s: (b') ;. ; (b(n-1))

/

(s .4 .2 .7)

;;

,_,, ro!,l

li.r, ra(")r-l

appl

(b) +s , g- s , (h) * s , (d) * C , t j J - g , (k ) +g , . ,t(d)*r, tr r ( b ) * s , ( e ) ; ' (j)-s, (k)*s, (b)*g, (h)*g'B+E'DrF'(d)*s, (f); x),

, and |lt*e*u-o-.t-xl l'

+mr+ ' +m

We rnay

111

Multiple HYPergeometric Function

|

I (,)**,*.*' n_r, (bt"rf'1i_ [-1.1*n,,

Re((a)-s),Re((b')-s),Re(s) > 0.

A=C=0,

Sec. 5.4.3

-s:

-1-^ +L.ai Re((a)-s),Re((U(n)) -s),Re(s)

(s.4. 3. s) , 0.

other lr'le11in integral relations involving hypergeometric functions of several variables exist. These results have not, far, been worked out.

Many

so

5.4.4 Special Cases

(

(c),m1*.**.,_l) ( (dr),ml)

"

.

(

(d(n-1) ) ,m,.,_r)m,

[-r,l * *r*'

r*rFc*o

| 6s1 * tl*' 'l'hc hypcrgcometric functio nof in1'cgra1'repr:esentation: -

\:

r

function of (5.4'1.5) may be so a large number of Me11in deduce to us to enable specialised as integrals of the special functions. Hence, we have

The generalised hypergeometric

.mn- -l _t

l

(s .4 .3 .2)

(5.4.3.2) has the following

;p2x/ 4)dx 7*'-t ,.(p/x)dx = f;f* i*'*'/z-toFr(-;c+1 u 0

Barnes

)c ", l(s+c/2) P -

zct 1c12*t-s1

Re[s+c/2) >

o.

(5.4.4.1)

I

ItJ

'l'lii

Infinite

Integrals

lCh.

5

s gives a Me11in integral of a modif ied Bessel function.

Sec.

5.4.4]

ll9

Special Cases

Arrtrtirer example is

{.i-' .1"(a,br,.,bnid1,.,d ;*r,.,xn)dx,

. s-l y(a,pxJdx = -p -s-a sl(s+a), R6(s+a),Re(s) > O, {5.4.4.2) Jx : whereby an integral of an incomplete gamma function is evaluated 'l'lrc integral

- (a-s'Dl .,t h ta-s,bn-sis,dnlp(n-l),. = f'Idn-r,"a, '''"n-I'"1 bn "Jtso Re (a-s) , Re(br.,-s) , Re(s) > O,

u

(s.4.4.3) l r'-' exp(l/z) \"(p/z)dz, involving a Struve function, may be evaluated as a specj.al case of (5.4.1.7) by expressing the integrand in terms of a ,F, series. Ilence, {5.4.4.3) takes the form pt/2*1r (l-s-c/2) , ,r-s-c/2,1t _^2,n, (5.4.4.4) y t,), ffy'n2-t(c+3/2) rlr,./) z/)tn/2', l"u'rZ'i'1, o"t, _s_c/2.1 > o, and this, in turn, yields the two ;i;"t.t /Z p"' = /r=-l "*p (t/zt Il t r2(ptTldz f=Illll-s) ,'rru!1)' ,-p2/q, --)/t' 0 -i2 , larg zl, Re(5/4-s) > o (s.4 .4. s)

'0

and

s-i --[n) j., - r;'' (ar,.,an,b,,.,br;c;x,

T

-

J" i-'

a["'

(ar.,-s)

,

Re

(bn-s)

,

(s.4.4.

e)

,.,x 'n-nldx

= f rl_:,u";,;;,.r.["-t,,,r, .,an_1,b1, Re

,dn_1ix1,.,xn:i),

Re

(s.,1

>

, bn _

,

;c

- s ; x1

, . , xn

o,

_1

)

,

(s .4 .4 . 10)

(a, b; d, , . ,dnix1 , . , xn) dx',

0

,dr,_li*1, . ,*.,_1) ,

(5.4.4.11)

o

t--s/4 exp(1/z) A_17:e/z)dz = lz p3/4 (s .4.4.6) "*p(-p2 /q, rl2 > larg zl We now give..two integrals each involving an Appe11 function, by special islng (5 .4.2.5) . )L

0

/rt'r 0

p, '

(a,b,b,;c;x,y)dy =

rr'.];l';],i;tlrrrt';l;b:*l -5r dr

g.4.4.7)

and

i "-r lrt'' 0

o

r(a,b,b' ;c,c' ;x,y)dy = rt';:l::;]'i;o']rn,

{'-t;b;*). (s.4 .4.8)

functions F, atd F, have recently arisen in a number of applications. See Exton (1976) Chapters 7 and 8 for example.

The

Equation (5.4.3.5) yields !i,tet1in integrals of the four Lauricella functions r["), atn), t[") ,na rjn), see Lauricella (189J). These results are now given.

l;c-s;xl,

.

,*.,_l) ,

(5.

4.4.12)

Re(a-s), Re(brr-s), Re(s) > O. In the expressions (5.4.4.9) to (5.4.4.I2), we have the additional restriction that n > larg *nl. This concluded the discussion of infinj-te integrals.

t2L

Generalised Hypergeometric Function

Sec.6.l.1l

Hence, the integral (6. I. f) becomes r 1ar) r

Cltapter 6

(br-ar)

f (an) f fbn-an)

--lCJ-

r(b1)

Multiple Integrals

provided that

6.1 MULTIPLE EULER INTEGRALS

Intcgrals of this type are generalisations of the single EuIer intcgrals discussed in Chapter 1. We consider the two general l"orms which follow:I I , n' tiul"'unro ,bn-'r-lf(u,,.,u-)dur..dur., ' i.,,,,.i"ir-'1t-u,)bi-ul-''.rin-'11-,.'n) 0 o (6.1.1)

Re (a-.

)

, Re

(b. -a.

.(rr,r, ) /{nl--,rJ

. . (an,mn)

-(u; \ .-\,

) >

^

tr m,, ., m n' (6. r .6)

O.

If (6.1.2) is .treated in a -similarw-ay, it takes the fornt r(ar). .r(an)l(c-a,-.-rn) .(rt,*11.. (ar,mn).

ff)rcoffi,r,,,n'(6'I'7)

provided that Ite(a.), Re(c-ar-.-u.,) , O. The forrnulae (6.1.6) anC (6.1.7) are used as a basis for deducing a number of results involving various types of hypergeonetlic functions. See hthittaker and l{atson (1952) page 258

and

/

rul

./ri,-'..rln-' (I rr-.-un)t-'l-'-an-1ftu,,',u,,)d'l

6.1.1 Generalised Hypergeometric Fuqction

.du n'

(6.1.2)

where R is the region g A ul,.,0 1ur.,, I : ul+.+un. It is supposed that the function f(ur,. ,rr,) takes the forrn

f(ur, ,rr,)

=

-m-m ) A *r,.lrn=o'l''''n

u.I l..u n n,

(6.

1

.3)

Amr'''m, being an arbitrary coefficient independcnt of the u.t . !',tany other types of rnultiple Euler integral exist, but they are not considerei "xplicitly here as their theory may be o

(6.2.2.1)

and

(o*)

same methods. 6.2.

;

*,,

':'::l]'*,.. I a,,-.a,n ; (g'"');ur'''ur",J

results of the previous two sections may be further special ised, and of the veri- many possibili.ties which may arise, we gi-ve the expressions be1ow.

= r (ar)

A number

.

.

The

Similarly, if we ernploy Hankelts formula for the gamma function, wc have the result r(or) ,(o+') )'- ' . (n).J'- '"uI*-*r, ,rol..r.,-bn f (r; ,.,r-lldrl..drn

= r/[l(b1)..r(bn)r;Amr .,rrr[(br,*r1..(bn,m,.,)1.

.

An integral formula and (6 .2.1 .2) 6.2.2 Special

Re(ar),.,Re(a ) >

.

;

=

trl_jto " t*'*'n rr;bi..r;b., rFo(-N,d;-,fi..*\ar,..a,-,n In

t/[r(br)..r{urr)]rI")(-N,d;b,,,,bnixl,.,xn).

(6.2.2.2)

The first cf these two formulae may be expressed in terms of the Laguerre polynomial, and the second in terms of the Hermite polynomial. If we consj-der examples of integrals of multiple hypergeometrjc functions, thcn we have -' -ur, rfr -'. .rlrr- Ivjn) c,, .,.n *rur, .,x,.,un)du, . .dun /. t"l . /e-u1 1u i i

oo'

= r(ar)..r(ar.,)nf") tu,ur,.,aric1,.,cn;xl,.,*r,), Re(ar) > o (6.2.2.3)

-,m Multiple Integrals

t26

lch.6

Multiple Barnbs Integrals

Sec.6.3l

If, Re(a.+b.) > O, I.S i S n, then each of the integrals of (6.5.5) rnay be evaluated by' means of Titchmarsh's formula (4 .2.2) , so that the integral (6.3.I) may be written. as

and'

' *1 *rr. 1(o*)"'I*'*'., u-ul. .r-"n ojn) r,, '''ni"'\:''rr.r' /(o+). (n) . 'ln .' x d', ' 'd'r,

d;

I B. ( [.,*b., ]/2,n) ( [^t*bt +lf /2,n)

m=o"tII

.

[r("t).

1/

.r1arr)i6F1 (-;c;xr+.+x

'

(6.2.2.4)

J

127

If

we now

6.3 MULTIPLE BAR}.{ES INTEGRAIS

1

1ar,+brrl

/2,n) ([arr+brr+1 | /2,n)

zn.

(6. 3.6)

take Arr(ur,.,rr,) to be of the form

The integrals under consideration here are of the two types (nl

-

i

cr+i-

c'

'[email protected]

.rrt ./ f.,l (2ni)n'c'-ic''-16

r(ar+ur)r(br-ur)..r(an+u,.,; l(b 'n -un'l .

(6. 5. 1)

and

cr+i-

-

(2ri) " -*t

c'-i-

.(")-/

:r((b(n);*u.,;

-i* r((f) +ur+.*un) r ( (g')*ur) . "(n)

.

r 11g(n) 1*u,.,;

x r(-Lll)..tf -"rl(-xr.)ur. [-Ir.,)un f (rt, ,ur.,)du, ..drn, {6-3-2) where the.quantities ct,.,c(n) ,r" real and non-negative. The function f(ur,.,lr,) tu. expanded as the series f (ur,.,u,.,)

llour!"r, lrn),

(6 3

3)

where the coefficient A*(u1l'.]rrrr) is to be specified, depenrling upon the type of integril inder"consideration.

is given by A*(ur, .,lrrr) = (al*rl ,m) (br-u.m).. (a +u ,m) (bn-un,n) *,,B -m' (6.3.4)

Suppose t.hat An{uI , . ,rn)

and make use of the multiple Barnes integral representation of the generalised Kanp€ de Fdriet function (5-2.3.6); we may show that the integral (6.3.2) is equ?l to the e-xpression

f-(a): (b');.; (u('));. ,l i , ((u),*) ((u'),tn).. ((u(n)),*)rTnr:n t'a| (f): (g,);.; (c(")),^I"'^ti *10"* )' L (6.3. 8) .

-]

of results of jnterest where multiple Barnes type integrals of numerous types of hypergeometric functions maybe deduced form the eipressions {6.5.6) and:(6.3.8) if the function f(u-) is suitably specialised. It must be stressed.that other multiple Barnes integral formulae similar to the aboue expressions may be obtained bi sinilar means, for examp1.e, if (6.3.5) is presented as a multiple series. Finally, we poinl out that cert'ain multiple integrals related.to those discussed above may be deduced using the formula (4.,2.4.1) in place of (4.2.2) above. A number

is an arbit.rary coefficient independent of u1,.,u, andz. m integral (6-3.1) then takes the form

where B*

The 6

I^

m=u

Brr*

.

#l

Ct+i-

r(ar+m+ur) r(bt+m-ul)dul. . .

c'[email protected]

.(n)*1-

x-, Ir 2ni) c' tn) -

f (an+m+urr) r (br.,+m-un) dun

(6.3.s)

[email protected]

*.t-P

F

-\

Generalised In{inite Distribution

Sec. 7.1 .21

-1

+(r) Clraplcr

7

in the

Applications

O

rlistributions of onc variate may be expressed as combinations o{' hypergeometric functions, and the density functions, characteristic functions and cumulative distribution functions of mathematical. statistics then involve integrals of hypergeometric funCtions. In this section, we examine a few of the properties of two distr.ibutions which generalise, respectively, the well-known beta and gamrna distributions associated with many classical problems in stat i stics. 7.I.1 A Generalised Finite Distribution We consider the farnily of distributions which have the probability density function [-- rnt- iIa') ;n,*1... 1(x.1 - ,. 6*d-I R,Fn, _F..1(a"l;h l_iU,) ; 1-.J AtnJ Btn)[1utnJ); (7.1. r. 1) {'(x) = O el sewhere . A'+l a u',..,A(n)*t l- B(n) ' a is real and positive and the modul i of the quantities h.,,.,h. are ali less than or equal to unity. 'l'lrc constant K is to be'determined so that

'.1'o:=i:x'

/

r1x1

I

ax

=ir(*)dx=l

(7

.1.r.2)

0

It will be seen that if A!=1 ,Br=O and n=l, then the density function under consideration is that of the beta distribution. Hence,

*-' - i: *u-'o,rr,[8:];rrI .;r(n)l- o ,1u,, = 4-1 ,1:A' ; 1:Br; .;s(n)fo*r,io,1

o

r,r

F,

i"r[

. . [:"]l

;;1"1 ]

:

, ,j

. ;", ,'i

o,.

f 1*1 d*

(n).

ir,r[r.,,_:[ ]; ;[,,;]

L:O;B';

distribution F(x) is given

r(x) = /f1t) at

(7 .7 .1 .4)

=

/r1t;

,,,n,,

I

.(7.1.1.s)

,n,1

by

at

I (7 ,I:A';.;q(') | a ,gu,);.;(r(')). i r 6) ,n"1 i';,;,; ,,,",[;,,i; ;; ;;;,",],n,*,

K *d =o

If we put x equal to unity, then F(x) = l, as would be expected from statistical considerations. The largest order statistic and the varlous moments of the distribution characterised by (7.1.1.1) may readily be deduced.from the above results. Infinite Distribution

7.1.2 A Generalsed

Similar expressions ma ybe obtained for :he generalised distribution whose den si.ty 1S -D-\ x-.'O,FO, d-1 [(r') ;r.,-J.. t(x1 = .. 11" ' l,'(n')

gam:ra

.r l,O(n), l(u') ; I .l "o(n)'utn)

A' 1 B',.,n(n) .= B(n) ,rd d and p are real and posltivc. give the followlng results without proof:

where We

(4).i:A'; *-1 _ r

po o:Bt;

;:::lt',

[:]:

(7

.r.2.2)

::::l l;n,,v,n.,0], h, o(t) = {' -U!)+r :A';' ;Af"]l o'(a');.'(,:']1, ...5 (p- it)o o,B',.,u('l[ -, io'; ; ; it('); ;p-lr"'-r']'(7't'2'3)

'if-,

See Karlsson

The

:

=riO-, * 11:o;A';';A(n)la '-;(a');.;(a(n));--. (p*)d

(7.1 . 1.3)

characteristic function, denoted by 0(t), may be represented itx as -l'

I.NIT

v(r,e)

Let us impose the boundary condition ,, 1r, cos-11.,; + f(u) as r +

n) =

/(n+1

/4

_{zF t(-n,n+1 See Sne c, . (7.9.6) l'

now defined by the equations

r(x)

fFt(x) =

|

frrrx)

os15ol x > cl,

(7 .s

?.IO THE LINEAR FLOW OF HEAT IN AN ANISOTROPIC SOLID MOVING IN A CONDUCf,ING MEDIUM

The problem of conduction of heat in anisotropic materials has galned nuch interest recently. Problems of this type occur mainly ln wood technology, soil ruechanics and the mechanics of solids of

flbrous structure

In this section, we shalt.consider the linear heat flow in a finItc rolld with conductivity K=Ko(l-x2), and moving in a conducting nodlun with constant velocity. Saxena and Nagara (f974) have recantly discussed this problem with the help of integrals involving Jecobi polynoniats. Suppose that the solid rod -1 : x -1 1 is novln3 elong the direction of its length. The flux vector is given by

(/.ru.1J I = -K grad u' function position and is a of and time, whoro u ls the temperature For onc-dimensio:ral flow, the single component of the flux vector rlonS any plane at a distance x from the origin is given by f=-K!g*p"rn {7 .ro.2) HtrG, wo have assumed that'the solid is moving with a constant valoclty v along the direction of the x-axis. Also, p and c are thr donalty and the specific heat of the solid resPcctively.Both thata qurntlties are assumed to be constant in this study. Hence, by rn rppoal to the law of continuity and the fundamental laws of htft trrnlfer, we arrive at the following differential equation conduction: of hcrt -iotii-;;**r

-

pcv/r

fi

+

Q(x)/r = rclr S,

tr?_ffir7t]T-l.f

-Tr)-r-

(7 . 10. 3)

r

7r'rffirw

143

with the law of conductivity K = r(1-x21. Q(x) is the intensityof a continuous source of heat situated inside the solid. Let the initial temperature of the rod be given by (7.10.4) u(x,o) = C(x) Equation (7.10.5) is easily conparable with the Jacobi equation

(I-*Z)y" +

.7)

Hhtnc0 F(x) ts the Fourier cosine transforrn of f(t) and so f(r) lf thc Fourler cosine transforrn of F(x). This gives the solution of thc problom in questj.on together with (7.9.7), {7.9.4) and (7.9.5), Integral eQuations of this type occur when problens leadin3 to-the separation of Laplace's equiti.on in toroidal coordinrtlr rro considered. See Sneddon (1972) page 4o7.

Flow of Heat in a AniSotropic Solid

Sec. 7.101

[(B-oJ [email protected]+B+Z)xlyr + n(n+s+g+l)y =

6,

(7.10.:5),

which has the Jacobi polynomials

o(*,8) (*) = {g**2F, (-n,'+l+B+n;a+1;13') as.

its solution. If

,

=

g:q,

Q(x) =

(7. 10.6)

we take

-(o+B)ix*,

q = P,

then the solution of (7.10.3) can be written in the form

.u = J^-{ne-Bnt o(o,B) 1x;. n=u Substituting this into equation (7.10.3), we have

(7.1O-7)

Bnq= + (n+c+g+t) In order to find the value of .{r.,, we make use of the initial condition (7.t0.4). This gives _

(7.10.8) r(o'B)1x;. e(x) = "lo\ by (1-x)aii+xlQt(o'0) (x), and Multiplying both sides of (7.10.8) integrating with respect to x from -1 to 1, we obtain the re:ultl

Ann'n' = G 76(c'B)

where

c

=

_irr-*)o(t*x1u

o,l"

B)

(7 10. e) "

(*) e(x) dx.

(7.10.r0)

This is because the Jacobi polynonials possess the orthogonality property , I i(r-*;"(1*x)B pjo'B) (*)0,1"'6) 1*1 d* = o, mln, -1

= ol",B), *=r, (7.10.11) where

6

(o,

B) = z"lB*l-.rIn*glI)r(.,*q*l) n ! (o+$+2n+1) I (n+o+$+1)

t7.LO.12)

-

-'!

l44

lch.7

APPlications

Ilcncc, the solution of the problem

may

be expressed

(7.10.13)

n=(]

iIIustration, suppose that g(x) = 1+x and v = l.Further, - l/2,8 = -l/2. Substituting these values into (7. 10.1C) irrrrl rrs ing the result

Ily wuv of wc 1:rltc c\

I

/(r-*)"(t**)B*k pl"'u)(*) = o for o 1 k s, fl, -r ror k = n, =

':'u

,

wll('t'(!

w0

u

a,B

,)'+cr+B+n, (1+cr+n) I (1+B+n)

u= n

r/2

forn=O

1t/4

fol n =

1

O

for n ?

2.

(o.B) - PI"'B) (x) = Now,P)-'-'(x)-iand U (7.1O. I3) gives u(x,t) =

[(o+B+2)

(7. 10. 14)

may be

(7 . 10. 16)

2 d2a* oosln 8= u'

(7.10.17)

are orthogonal over thc interval I -1,1r-l with l/2, and they nay also be respect tothe weight function (r-*2)dobtained from the Rodrigues formula

r*l = ol") (l-*2) -d'+1/2 r$;l'cr

,"h"." A(o) i s a

normal

-*2rn+a-r/2

isat ion Factor given

, 0.tI.4)

by

(o) (-I)nf (ct+l/2)r(n+2c) A ^ n = 2n nlr(2o) r(n+0+1/2)

(7.11.5)

AS

(o) ^ ','n

(-1)n 2n n: (

2n)

(7.11.6)

I

This is consistent with

cjo)f*l = Tn(x) = cos(n co=-1x),

{7 . rr .7)

Tis the Tchebycheff polynomial of the first kind. Other n sets of ultraspherical (Gegenbauer) polrlomials are the Legendre polynomials, for which c, =7/2, and the Tcheby,cheff polynornials of the second, for which s =1. The Taylor series expansion of an analytic function about the origin corresponds to its expansion in uiclasphericai polynomials for which c + o.

(7.11.1)

ultraspherical polynomials on the interval [-A,A] ar:e defined as the sets of polynomiais orthogonal on this interval wlth respect to the weight function (t-*2/x2)a-I/2, where a > -l/2. The normalisation is chosen to give rise to the polynom;ats C{o) (x/A). Approximating sin 0 on the interval [-A,A] with the ultraspherical polynonial Iinearly, one obtains The

so that one obtains the approximate solution (7 . tt .2)

the approximate freouency, is to be determined. Aquite gcneral approach to this problem may be made by carrying out a I inearisation of the term sin 0 using ultraspherical poll'nomials,

t-

.3)

where

dr7

whc::e or*,

11

These polynomials

x+(a-B)l/2, so that

written

0* = A sin(o1t +6),

(7.

r^ I rn) Then, Cf *'(x) = 2ox and C'"'(x) = 0 fo:' n : I. Ho,rrever, in the case oF the Tchebycheff polynomlals of the first kind from (7.11.,1) with o = 0, the normallsing factor must be re-defined

ln the investigation of certain types of non-Iinear osciliations, a linear approximation which girres the greatest accuracy, in some scnse, is made to tire non-linear ternrs of the governing di-ffercntial equation. This dpproximation is made over some interval [-A,A] , say, of the dependent vari.able. This process, exemplified here by considering the free oscillations of the simple pendulum, enables us to compare the solution of the approximate differentj"al equation with the exact solution of the original nonlinear differential equation, of motion

t45

(2o,n) . . .(o)..., .-.1-1-x.),rr-2. I Ln-txJ -= -nl-Zht[-n,n+10;0+2; 2

cj")

7.I I AN APPLICATION IN THE STUDY OF NON.LINEAR OSCILLATIONS

The equation

Study of Non-Linear Oscillations

(7 . 10. 1s)

I (2+c1+$'t)11)

gct the result

]

given by

AS

rr(x,r:) = i^a,,,0,1"'u')-1"*p[-f(n*o*o*t)]P,\t'u) (*).

Sec. 7.1 I

where o

'in*o = tl")c(")

(o/A)

'

,f") ,-n ='j6-02/n21"-r/2 r(a)(0/A) sine ds x

(7.r1.8) (7. 11.9) (continued')

l,ltr (

Applications

lch.

7

t'orrI i ntrerl)

fn

"ll

L_A

ttlr

1r;fo)ro,ral 12 t1-02,,A,2)'-t/z

ri ',

(7.11.e)

Sec.

7.13]

so

that

ai'

= l(cr+2)Jo*, (A)/ lcrin/2)ol ,

s() that the required linear: approximation

(7

of sin

.tL ro)

0 becomes

sin*6 = t(o+2)Jro+1(A)o/ (A/2)o*l = Ac*t(A)0, A^(A) (A)/(A/4o 0-- = r(o+l)J^o.'-

(7

INTEGRALS

A molecular

integral of frequent occurrence in quantum chemistry is the function Fn(z) given by n=O,t,2,..

(7.12.t)

See Abramowitz and Stegun (1965) page 228.

It is of interest to develop the Laplace transform of the product of several such functions, that is

j"-ot - ' - (at ,. , u-) . = o ,' ' urr,'rtr"Bnr(a.s)ds, ",r,nr,.,nrtot where p Il 'P-

p and are both positive. By an eLementar:y change'l;'?t;3' variable of integration of the right-hand member of (2.12.1), it may be shown that (Abramowitz and Stegun [1965] page 23O) Brr(z) = r-n-l[r(n+1,-z) Now,

- f(n+1,21].

(7 '12'5)

-r

(nj +t ;n'+2;-ajs) -tFt ('j +7 ;n'+2:a's)lds {"-ot and this integral may be evaluated b1- using the formula(7'12'6) o1 l^" " r"'tFl(bt;c, ;xrs)..lFt (b.;cr;x.s)ds

= r(a)rjt)fu,bt,.,b.;cr,.,cr;x,,.,xr)

.t .t2)

7.I2THE LAPLACE TRANSFORTI OF THE PRODUCT OF SEVERAL MOLECULAR

dt,

(n*l)-lIrrrIn*t ;n+2;-r)-tFt b+ttn*2;z)f.

0

It was found by comparison with the exact solution of (7.11. 1) in tcrms of elliptical integrals, that a good approximation to the solution for A in the closed interval [O,r] is obtained when u = -0.21. This method gives an amplitude;dependent solution to the problem under consideration which accords very well with rcality. For more details see Denman and Howard (1964). An obvious extension is to a stuo

I'

4.1.1.13

(-

-T(a*U;

2',

r(a)r(b) - Irfi*tl-- r''z - -la+b

,OFt(-;b;u[1-u]x) [ 1-u.)

h-]

"oFt(-;c;u[1-u]x)

ua-l

.-

ilrr;n

',

a+b a+b+1

or-I

2 '''

I x. (-; a*7;4)

a>O

a

I

a,b > O

It- 2 '

a+brl 'al 4l

'i

2

r(a)r(b) - J-, u ;l 2''3la*b a*tr*l

lG-bl

l-;-

a,b'O

.b-I

1_-

,

^.1I

'?

'L,e

)

I

xrF, (a+b;a;ux)

+A+P

"*

-- . b-1 ua-l [I-u.) xrFr(c;a;ux)

iffi

t(a)rIb)

rFr(c;a*b;x) a'b > o

.- .b-1 ua-l tr-uj

r Ia) I (b)

tr- tl

[1-uJ

I (a+bJ

x,F, (c;d;ux)

a,b > o

la , c:x "[_a*b,d;

I I

a,b>o

_l

a-1

1

*OFt(-;c;ux)

Z'

-- -b-I ua-L (r-u]

a,b>o uu-

rya+U] a,b>0

c;u-x)

f(alf(al

u A.1.1.

;

du

a a+l 2' 2

r (a) r (b)

,'-I(1-r.,)'-I *oFt (-;a;u[1-u]x)

c-l

A.1.1.9

OF EULTR TYPE

') ,OFt (-

f(u)

['o

1"

2

[- -T a*t

a>O

I ;*l

Ia*O a+b+I

lr' I

z ;) I

--a+l-a2 x,F,( . iI+^;u IlZZ u "-'Ir-,r; x

t

2 - -a a+b+l ;ux)

1r1(7; 2

ua-l A.1.1.14

o-

x_l

-- -uJ-b-l (r t)

xrF, (c;|;u-x)

ex,/a I (a) I (b)

a >0

,a+I _ a+t

ffiii'rr/f;|;n

a'b > o

. [+,. ,lx *t a,b>o l-', ^*t 'l

r(a)r(b)

J-(a*D-2' 2l u1a

I

lntegrals of Euler Type

154

lA.l

A.r.ll

Single Hypergeometric Functions

155

1

f (u)

1l

-- - b-I ua-I (r-u) A.

i,l

.15

')

x,F; (c;d;u-xJ i. t-

,-ro1l-rrlu-t A.1.1. t6

4.1.1.17

A.1.1.19

ua--(r.-uJ-a-l xrF, (b;a;u[1-u]x)

xrF, (c;d;u[1*u]x)

ua-l

a>o A.1.1.25

r(a)l(b) . f-c, a, b ;;l r 1r. U1-S"3 j , a*b ,, trt I .*l J-"' : ' 2 ''l a,b'o

c+d+l xrF,(c;d;-f;u)

,|

.- - a-c uc-1 (I-uJ ux a; c ;r) ,l; ,la ,

-,r-r: u (r-u] xr!,(tr,1-b;c;u)

r

_I_c_d r(:j:)^,, r('-jj ,

+a

a+b^. ( 2t 1 ' a*b > -1

-- ,-b-1 ua--l (I-uJ _ ,l+a/2, I 1' a/2

-

>

O

21-2a

a-c >

A. 1.1 .29

a

f

-1,-2,.

E (q h'c'rr/?l

2' 1 '*',

C>o,ulb-.r-t

*.,

{o,-r,-2,..

^1 -4-! ( I-uJ uL-1.-rF ,(a,I ; c; -u)

n

r(Trr(!+1+a)r(l:f.) !.1^

(c- 1)

(a-1) (b-1) 2b(a+b+l)N! (b-a+I,N) (b-a- l) (2b,N) (a+b+2, N)

h(h/2-1) (h+k) h,h+2k > 2

@f(a+b+N) (l-b,N) a,b > O, N=0,1,2,..

r (c) I

(+

-c.+t)2/n

(a-b)

*[{r(5r(tE) }-1 -{r1!er.fL,

f

-1t

-.1

r(a)r(a-c*t)r(c)

xllr(l *a- f)

(a-1) (b-1)

a,b > O, N = O,1,2,.. ,-rt-1(]-r) a/2+b/2-c

f (c) f (a-c+1) -a- I _ ---T-G.Tt-

(c-2,2) r (c-2) r (c-a-p+4

(a-2 ,2) (b-2 ,2) I (c-a) r (c-b) (c- 1) (c-2,2)

r (b+N) r (a) r (a+b-c+N) I (b-c) I (a+b+N) f (a-b-c) I (b-c+N)

A.1.1.28

2a-c-d > -l

"t,r,ffl -,t'(?)l

>-1

2-2

xrF, (c;b;u[1-u]x)

du

(c-1) r (c-1 ) I (c-a-b+1) (a-1) (b-1) I (c-a) r (c-b)

lt-uJ-h/ - , .I-k,h/2+l; ,,. l'1\ h+k ; "' A. t.L.27

-2a tfr)

a,b I \, c-a-b

f(u)

{a-2,2) (b-2,2)

,F,(a,b;c;u)

a>0

ifi'-,F,(h;^.!;|t

"ii

c

4.1.1,,,,,

c-a-D > -l

A.1.7.24

a > 0,

A.1.1.21

x/4

all,2.;bll,2;

A.L.1.23

'',

-- -ir-l (-t-uJ

ua-l.-( 1-uJ- a-l 4.1.1.20

lfa+b) " "l a+b a*b+ I ;. i-2' a,b>O 2'"'i

Ifa+1)f(a)

(l-u)rFr(a,b;c;u)

. ;l xl

,|i,*, 3'3l' .'

r(a)r(b)

'r1., (a+|;a;u[1-u]x)

,'-11t-r;b-1 A:1,1.18

'r';

Io

o'

a > o,a-c

-- - a-b-c uc-1 (r-u]

>

A.

1.1.3r

.rFr(a,b;c;-u)

l(l)$.t!,rrE,

t-{,(cr}

c>0ra-c>-1

i(c)r(1+a-b-c)

I [ 1+a) r (l+a/ 2-b)

c>O,a-b-c>l,b>1

"I t56

Integrals of Euler Type

[A.l

I:

f (u)

-- . a-b-c+I uc-i (r-uJ x^F, fa,b:c:-u) I L' c > 0, a-b-c > -1,

b l/2

t

rG)_ | (d+e)

lrdl

'.1F=

a

1r,b,d,c; I d+e d+e+ ,

f'r' ,'

I

x/ al

)

d-l (--l-ul . e-l r (d) r (e) u I (d+e) (- a,b, c;u [1-u]x) "OF: ;

""'lo.l'01".,

'.'*-l L-z-'---r'a'D'c; J

.- - e-l ud-1 tr-uj (a,b;d;u[1-u]x) ,F,

d,e > o

r (b) /n 1

X

__l

d,e > o lxl .

r(a+b) 2x(I-a)

;1

1

ll*2b s*zt "rr rl-r. ',---t-t *r b>O

b

b-

u-l/2+3a (r-6,r^. f,

r (a) r (b)

t*u*t. x^F-l 2' 2'i"u-l,

(a+b)

't

d,e > o

b-

[-91q

,A.1.1.56

I

I-

A.1.1.60

A.1.r.61

"srfri,rT',ru)

du

r(tr -"lr(a+ ,| - 2,O l*l

L

_l

t*,tt . +

-

,l |

I (,{)

Integrals of Euler Type

lA.l

Single Hypergeometric Functions

A.1.11

/l tr,1 o,

f (u)

.- - b-l ua-L Ir-u]

{ffir,-,/D-b A.

a,b > o -ua-I [Iu.)

4.1.1.65

].1.70

-1

r (a)

.T

A.1.1.71

t -,1

4u( r-"1 .

A 1.1.66

-1

2

LA

-t 4u(1 _ul

;

I

J

f(a+51'

A.I.1.67

1

i 1r -r1b-

I-a+3b-l

"2Frl 2 ' La

2

au(t-u!

,'-11t-r1b-1 A.1.1.68

au[1-u]) 3a-5/2

4,1.1.69

e_qa-r/2 la,2a-l / 2; trl t ; L5a-5/

au(r -u)

!

r(r-a-3b)

r(a+b)r(a+t)r(@a

/

A.1.1.73 2

"(+)

-2b

xrFr(a,l-a;1/2+b;

r(l

r {a) r (a* t / q r (a*

+2fi.(+.flrrf

xy/t(g/2-a/2+S/B)

[1-u] )

,rt-1(I -ql/2-c "rF, (a, I-a;c; .2 u s1n

s /,

q)

.Irt

b>

o

r(c) I' (3/2-c) sin [(2a-1) x]

2/r(2a-Ij ain a xJ

1

'..,t-

a > 0, b >

r (a+b) t

a,b>O

A.1.r.74

1

rrtlal rrTat ftla

-cl

r

1tE!;

0 r c < 3/2

A.1.1.75

l*l .

11-,r1

-l / 2-c

r (c) r

.2 x) u srn

,2 u srn

i.

1.76

cos (2ax)

O O A.1.1.91 b

b,c > O

,

A.1.1.92

-

{

a,b>o

O

,c

-r tr (q, r 13*?-zur,

- rr(+)re+4)1 Zlnf (d) r (c -d) r (.0-*21 ;u.

@

2(1a*b*2l/2,d;r)

x

d,c-d > O

^.... - ,a+l.a+b+l ,Frl , i-2-'cix)

(a*e) Z'

.

c-d- I

*3f^,a,b,c

(- ; {a+b+t) / 2 ;x)

I(c) (b-l)2a

d,c-d>O;bO

f[a)l(b)

un

+.B#p rr rta;\,etfl; ^r a)

oF1

ffibf

d,e >

1

less a(b)=6, -L,-2,

o

o

+A#P

,nr(f;b,c;u-x)

" 3F

"Ia*r )-l 'Oi

I

d,e >

r

ud-1 [1-- -uJ- c-d-1

;

. '*/al

r (d) r ie) )

ud-1(t-..r)" A.1. 1;90

l(d)l(e)

u* '(I:u) 2-.a+ba x1F2l 2;7,c;u xJ

A-^

I (c+d) 2r 3lc,+d. t_ crd > O L' I (d) I (e) r (d+el sFa

. e-1 ud-1-tl-u.)

oF,(-;c;x)

1

xrFr(a;b+I,b;u[1-u]x

ub-1 A.1.1.84

b-

xrFr(a;b,d;u[1-u]x)

I

7;

r (c) r (d)

o.,l

i

u2sIn2--l xl A.1.1;80

d-1

,d-11r-r1t-1

xJ u2s1n21

i1_,r1b-r

163

Single Hypergeometric Functions

A.l.rI

[ fla/

z) r {a/

2+L

/ 2) 1-

1

-[r(b/2)71a/2+r/2))

I

ln.t

Itttcgrnls of I')trlor'I'vPc

(r,l

.I I '0

f (u)

A.1.1.93

uc- I (L -uJ- c-c-l *.r, talb;.:r)

,",

f(rr)

Single Hypergeometric Functions

,t-111-r1 A.1.1.101

I (c)".-e, r (d-a) I (d-b)

e,c-e > O; d-a-b > O

=O,-l,-Z,, -- - a+b-c-d-N ud-1 (I-uJ A.1.1.94

,.r,

{r;b;-N

i,.,)

ub-1 I--I-u)-a-2b A.1.i.95

I ia+b-c-d-N+1) (c-a,N) (c-b,N ,N) t (a+b-c-N+1) (c,N) (c-a-b,N) a+b-c-d-N>-1 d>O

la,r*],-x; ,-F^l ul 5 lla ,-r* .'. I

'l

u--'l -(I-u)* -) -' - ,a,1+ a/ 2 ,b n

*rn/r;j;o-t -N ;..,

3'ztb/2,b/2.r/2;"1

a

t (c)t (r+a-Zc)t (r/2+a/2)T (t+T_bb) I ( 1 + a) t (l / 2+ a/ 2 -b) r (L / 2+ a / 2- c) ..t (r/2+a/2-b-c)

'ffi

;

_- - b-dud-1 (1-u) ,

-F ^ 5l'c,

(a ' i - 1 ' b

d

;

1

uz

z't

;

1(b-a,N)

r(t-l"' +5 rb.N)24lr

r (d) r

0

(c:d)l-("tl

".!,)

r'(.lr il/:l

)

r (c) f ( i +a/ 2-b) l'( I t rr ) O rtnloss

d(l))=O,--l,-2,..

and d,c-d > o A.

l.

ud-1,r-rrc-d-1 1.107

t

r' r(?'.3.^',/ j' i'*) b-c-l uc-1.[l-uj"

A.

a > -l

d,b-d >

a>

4.t.1.106

c'a-2c+l > o

._.-:---

c,a-2c+1 > O, and a-2' unless b=0, -1 ,-2, . . .

f(1+a-b+N) (-2b,N) b,a-2b+1 > O

rlIe-

A )a_A > / v6 u'-L-u

r (cJ i (1+a-2c) r (i+a-b) I (1+a) r (1+a-b-c)

3t'21 a/2, 1*a-b;-uJ

r (b) r (1+a-2F) (a-2b,N) (-b,N)

{,q)i.,1li;-N;"r

c

d-l c-d-1 u I--r-ui-

a-2b-2c > -1 unless b=0,-1,.

A,1.1.1

;

and a+b+2c < 1/2 unless

;u.

-- - a-2b ub-1 (I-u.)

b-l -)a .i)t" 51 | ()

3' 2\7 / 2* a/ 2nb / 2 ,d;u

b > o,a_2b >_i

A.1.1.105

"; rrii.,?13;3;"r

- ,a/ 2, b +N ,

r (d) r (2c-d) r (c+

l(l/2-a/2-b/2+c)

xl/i(b/2+d/2) e>O; 2c-d-e > -I and c > O unless c=O,-1,-2,..

;

[a*b*l )/2,d;T) uc-1 [r-- -uJ-a-2c

A.1.1.99

c

b,

c ?b-c-N+I > O

)

. '*r(1TI'b/T.a--'

.rd-I.1t-,.r;t-d-1

A.1.1-98

. - , a,

^

z1-2trr (") r (d) r (r*2.-d-.) l. d- -a+d-^ I *t -=r)|( r(;*; 2 )r(d-c- 7)

, _F^ (a ' 1-a ' c;.'.,-.}

-,a,b,c ^3r2 (

\-t- -'') x l'1+a/2-b.Nl /- /a r- rr\

(a/ 2+l /2-b ,N)

^ -C2b-r,N)-

e-1,(I-u.)- 2c-d-e u

A.1.1.97

i"l

-- .2c-d-1 ud-1 (r-uJ

2-l/2-b,N)

[_1

J l'd,e

.rn, ri;l1oN.

r(c) i:(r+2b-c-N) (a-2b,N) (-b,N) | (1+2b-N) (1+a-b,N) (-2b,N) ld/ z'u ,tr

(b) r (1+a-2b) (a-2b-1 ,N) (:b-1,N)

I

4.i.1.96

A.1.1.102

(d)

t (1+a-b+N) (a/

c,b-c > 0

-- - 2b-c-N uc-1 ll-u]

.

tr,1 o,

r (c) I (b-c) (a-b,N) (b-af (b+N) (1+a-b,N)

"r',r";))1";-*;".,

unless a(b)

165

/l

f (u)

du

t,a*,

*,

A.l.1l

t. L to8

*

,,

,(

1)^/.r*

l,o i".r

ulr L{s--O l:

-

r(c) [1+,/(l-x)]24

q.-{-3_o_ lxl .

r (c) r (b-c) z2u-r

t

:-----Tf6l-diij 1t*/1t-*) c,b-c > O lxl .

t

I

1-2'

lntogrnlr of Eulor Typo

t66

lA.t

,c- I 11-,r1b-c-

t.l

.

t09 "

1

iiff#hgE

rr r(Z)l/,'*u;b i,n2)

,c-I 61-,r;b-"-I 4.1.I.110 " r, rri|liao i-u*',

-r ( 1+x)

A.1.1.113

A.1.1.114

*

2

*;

,rrr{a;b;c:rx)

1 1,/1 r +x21

1t

-r;

I

xoF, ( -; a, b, c; ux)

A.1.1.117

"oFs(-;a,b,c;ux) -- - c-1 ub-1 (I-uJ *oFs (- ; a,b, c;u [1 -u]x)

,..,t-1gt-r1 d-1

4.1.1.118

*oF3(- ; a,b, c;uI i-u]x)

1-

*7b

"

r(c)r(b-c) sin[(2a-1)x] r(b) (2a-1) sin x

A.1.1.121

c,b-c > 0 f(e)r(c-el -=tfr

l*l.t A.1.1.123

t

f (e+

-r; 8-

A.1 .1 .724

A.1. i.125

rtb)r(c)

b+c b+c+1 x-

brc >

r0 (d,

+,si#1,,,

c,d > o

[email protected]'a;c'd'e+f ;x) e,f > 0

-

l'zl

ffifiunt r(-;c;x/4) u

[Lul

a,b>o

d1

-d+e

I'b;c'd'e;

lrrF=(a,b;c,d,e; uIt-u]x) e-1-- f-1 u

xrFr(a,b;c,d,f;

I

l1ffirnrtu;*|1, ; d,e >

r (d) r (e) I (d+e) ,F

c;x/4)

O

d+e d+e+I

r(a 'b; -7' d,e > o

,

x-

'ci 4)

l(e)l(f) - , e, a, b ;x. \e+f e*f*l -i (-ed) s'4 ,.6J ^ , ,-1-'c,di

e,f

>o

€_1 c_1 u- ^11-u1s '

4.1.1.126

2Fj(.,b;c,d,e; [1-u] x) -c-e-1 u

O

b

-fzF

>o

xrFr(a,b;c,d,e;ux)

u[1-u]x)

d,e > o

e,f

r(

[I-uJ

(d;d+e, a,b, c;x)

,Fr(a;c,d;x)

+o

ud-1 [1-- -uJ- e-1

TTfi.t- ot'3(-;a' 2 '--Z-;4)

{ffi,

I ,rf - 1t

l- c,a,b -- . e-l ud-1 (I-u.)

oF.(-;a'b'c+d;x) . c,d > o rFo

rFa

(f)

(a, b; c, d, e; ux)

[-a+b a+b+1 x -F t)' JI7t 1 ' ) L

rFr(a'b;d;x)

e,f > 0 l*l .

+ft*P

rl'!.j;"*l

-- . b-1 ua-l tl-u]

e,c-e>O

+?#P

e) r

I (e+f)

ue-1-tl-u]- f-1

4.1.1.120

o

_l

lA.l

Integrals of Euler Type

I (rt]

['fr'.r) 'o A.1.1.128

-- - c-d-I ud-1 [I-uJ *orr(X,r*|'-N,c;,,,

1'b' A.r.1.129

c

.ors (X,

?,1,-t",,

fe*l

>

r.. jt

*.F-[u'blt,dlux') 4 J'e,r,g I 1

*or. {a;b;c;d:,*) h_r i_i u" '(I-u)J *on, {r;b;.;d:r*)

r

ct#t A. I .1.138

c-d-1 ud-1..t1-uJ. x F ro,l+a/ 2,b, 5'4'a/2,1+a-b, -N,c ; ul,

-1

2*2b-N, d;

,d-l,t-rrc-d'1

A.1.1.133

, E rd+€ rl+-Tr, 4' 3'1*a*9,tiz*Ztz,

r(b) (l-x)' c,b-c > O

zF

r(i'*

A.

l.I.

139

-N,c ; . I+2b-N,d;ui

^7'.7

z

'")

d,c-d >

A.1.r.140

'roo

('ifiiilj

";

".,

,j-11t_,.,)k-I

0

('iliiilj '; ",.,

d,c-d >

,.p, (u;b:tid:tir*) ,)

0

r(j)r(e-j) -,a,b,c,d;., rGj--4'3t f,g,h;^i j,e-j,0 l*l.t I'(j)r(k) f(j+k)

p ra,b,c,d,e;_., 5'4'f,8,h,j*k;"'

j,k , o

,k-11t-,.,1 P-1 A.1.1.142

\r

0

r(:l) I{c-d) (a-2b,N) (:b,N) r(c) (1+a-b,N) (-2b,N)

.. - ,d,l+a/ 2 ,b , "5'4ta/2,1+a-b,

,j-161_r;e-j-1 r (c) r (b-c)

r(g)r(h) r ra,b,c,dt_i r (e*h) 4' 3 (e, f, g+h; ^'j s,h , o l*l . t

r(d)I(c-d) (b-a,N) I(c) (b,N)

5'4\b/2,b/2+1/2 b+N.-N.c:-Lrl-

,

"c-1,r-rrb-c-1 -a+ee

f(g)f(d-g) . .a,b,c;,.. --TTO s'2\ e,l ;nt lxl.t s,d-g,o

. r ra/2,a/2+I/2,

1

unless a(b,c)=O, -1, -- - a-c uc-I (I-u) I (c) I (l+a-c) (b-a ,N) I(1+a) (b,N) .. -,a/2,1/2+a/2,b+N, c,a-c > o ^q'srb/z,b/z+r/2,

du

r+a,d;,"

*. 11I -a-b +cl

2 "|e*l

f(u)

r(h) r(j ) . ,a,b,c,d,h; . -ff 5'4te,f,g,h+j;xJ , h,j o lxl . t

ud-I".[l-uj.c-d

-u,Nl

-u,Nl

r (e) f (d-e) r 1l1r

i.1.131

4.1.1.132

4.1.1.136

(2+2b-N) (1+a-b,N) (-2b-1,N)

A.1.1.137

-otr(ii,?i l'o',,l c-a/2-b/2

r (c) r(2+2b-c-N) (a-2b-1,N)

rtl

ue-111-r1d-e-1

and

A.1.1.134

f(1+a)f(1+a-5-s) d,1+a-c-d > O; a-2h-2c > -2 unless a(b,c)=0,-1, -2, :..

(-b-r,N)

1. 1t I+a-D t ci

. c,2b-c-N+2 > A

A.

u8-111-r1d'8-1

A. r.I.135

I'

Io

i h,rs- 1l _,r)

r (d) I (.1+a-c--d) r (t+a-b)

-ut ^ oF r(x' 1.7'.0'l 7, !+a-b,dl

I

f(u)

;

-- - a-c-d ud-l (r-uJ - a-

t69

Single Hypergeometric Functions

du

r(d) r(c-d) (a-b,N) (b-a-1,N f(cJ (b,N) (1+a*b,N) d,c-d > 0

-- - 1+2b-c-N uc-l (I-uJ A.1.1.I30

A.1:ll

,P

o

lxl .

r

Integr-als of Euler Type

l"to

lA.1

/f .c,; o,

1,,

-a-ze ( r-u] uc-1.-

u"e

.'6'5\a/2, c rz,l+a/2,b, l+a-b,I+a-c

,

A.1.I

1

1t -rr1

G'''

r

P,Q'

149

1+2a-b-c-e+N, -N; b+c+e-a-N,1+a+N;

rrx

l I

r (d)

8t,

".,T,

;iii[.1,1, ,,::,

ur.r'''''P or*r' " 'oJ

c,d > 0

(f) r (e-f) r ( 1+a-b) r (1+a-c) I (e) I (1+a) I (1+a-b-c) I (l+a-b-d) - ,a,7+a/2,b, c ' 6r Sl a'/ 2, I +a-b, I+a-c. ..f (l+a-d) f f I+a-b-c-d)

I

r

A.1.1.144

"

-- . d-1 uc-1 (r-u.)

f1t-a-cn)

'c|'I

f,e-f > O; a-b-c-d+l > 0 unless a(b,c,d)=O,-1,-2,..

* A. I . 1 . r4s

5\a/2, J +a-b. a+b, d, 1 -d ;.,. l*a-d,e;uJ

I

. Lr.,)

a+l/2=d/2,1+a+N tl

.147

i

-I d-1

Fr( (a);

(b)

If

p=q+1,

I (c+d) X

I

I I

I

I

I

!:.

-

-,an-.I-c r l-u: 1l

' ' -.bm'.1-c / 2-d/

I

p

1/2-c/2-d/2,b rilr

blql

I

I

Integrals involving Double Hypergeometric Functions

,1 J

.b-l ua-I-Ir-uJ

/6-d+N-1, d-a-t'r- >

r (c) r (d),

restriction on xifplq.Norestriction on x,P,q if an rat parameterl is a non-positive integer.

No

A.I.2 Euler

I+:-d,N) x(i+2a-2d,N)

A* I

rB*

^h!

O

'2F1(ii6-1q*11")

,c,(a)

1

tc*d, (b)

r-(al,m, c .

,'B+m+nlI -. A+n+n tthl I l u,

i*1.t.

l"r'

-n+lr""r o

A.1.2.1

;--. ;

"J

"rnur[!]:".t

c,d > 0

,l .1.148

Al.

(

r (c+d)

,RFe(8];"'II-u]nx)

A. I

r

r (112a-d) r (d-a-N) I (1+a+NJ

I (c) r (d)

,t-1(t-u)d-1 c,d > O.

-u] xl (b)

l_ c., d

a>O a-d+},a+d-e>O; unless a(b.d)=0, -I,-2,.. l

A.t.1

1

*

^ .n*zl-*l't'" *up*2',q*21

..f ( I +a -d1r Q/ 2) t (a/ 2) "T ( 1. a / 2 4 / 2 -dn) r(il 2.b/1- d / 4

6''

c-1-I I -!r

t

(a).

I (a) I ( I+a) i (a+b-d) I (2+a-b-d)

N-1 (1-u.)-d-au2a-d+N-- ,a,l+a/2,d/2, a-d, 61 5ta/ 2 ,l+a-d/ 2 , l+d, r/2+l/2.-Y ;.1

A.1.1.146

t"

I n-J /n2- " "

1.150

I(e) rIa+b-e) l(i+a-b) r(a+c)

- ,a,l *a/2,b,1-b,

t7t

Double Hypergeometric Functions

A.1.21

,-c+m-l x,

u

,

x

_c+d t 'm+n'

Ld C+n-II m n I -.''.--_ n' n mn x c+ct+m+n-1.(m+n)

I

I

m.n ' --nr*nI_l

|

A.L.2.2

f-1 (1-ul" -- - s-1

,r, rl:rl*., l"t

c|'Ir",.] [!]r f,g ,

0

o

f(u1

au

r (b+N) r (a) (a+b-c, N)

f (a+b+N) (b-c,N) - , (d),a,atb-c+.r-

"o'u(ilj

;;i;-; l'l-"1*)

a,b > O, OFU conrrergenl over the range of integration r ( f) r (g+N) r (g- a) r (f+g1 f (g-a+N )

*cp**^'n*? [*l r rl_l

dr

,q.

"1

1

+a-

f- g-N ,d n+l

l+a-f-9,

e

2,

t't

lA.l

tntegrals of Euler TYPe

)

1

/: f(u) u'-l(t- u )- -1 (:- b N +1 lu) '2F1 f) :(g ,,F:G;G 'H:K;K k h) :(k I

A.1.2.3

L(

the

-- -b-1 ua-I Ii.-u] ,.rnrr!:olr.ri,) (k)

"'

F+2 :G ; H+2 :K ;

;

(e )

-1

r'orl'

ii'i;"*'"i

>

range

ua-L.II-u]-b-1 ,.Fo( (c); (d)

A.1.2.5

._---_

'uF,.

[

(e)

K ;

;

ux)

|

;

A.1.2.10

uy)

>

4.1.2.6

,u

11-r'r1

r (a+b)

lq

(I-uJ- c-1 ua-1-"02(b,b';a;ux,uY)

A.1.2.11

/^\ -

A.1.2.12

range

T

*

I

A.1.2.14

n

I

:(d);

m+t]

-^- -ri-]

-m+n' (m+nj -m+nl (m+nj _l

f

I

o

Sa*l,b;

,.ru-Iit-r1

series cgt. over the

of integrat ion

1

b+n-l

a+b+rn+n-

'.x,Y)

ard >

l',!'.x/a,v/al ( r.,l r _l 0;

r

'

F

^l C^(b,b';a+c;x,y) :++l-:-!+u t' a,c > o I ta+cj

^^l

a>

m 'n'

Ii]: \r/'-

,.,111-,.r1o-'

(f);yu[1-u]l uFr((e);

a+m-1 b

l_m*n'

f

D;

d- 1 1t -r) * 02 (b , b ' ; c ; ux , uY)

,ru-

integrat ion

:

lm I a*b

'

'-

,aFr((c); (d);xu[1-u L.7.2.7

r(a)r(b )-m+n:C;E rm*n

O;

-- - b-1 r(a)r(b ua-l (r-u) r (a+b) x^F- f (c) : (d) :uxl L U' ur.((e); (f); [1-u]y) 3,b > O; serles cgt. over

a+b+m_1.

(c); (e); ,l (d); (f);^'Y ) over the range of i-ntegraticn

a,b > 0;series cgt over the range of int egrat ion

i

I

.

-b-l ua-1-i1-uJ xaFr((c);(d);xu* [r-"] ") xuFa((e); (f);yu '[r-.'.,]')

seri es cg t. over range of int egrat 10n. 3,b

:

I

'l(h) ,a*b- c, a+b+N: (g') ; -. .-

Lt*u

m:U:l-l ...."'.la+b

m"m

a,b > O; series cgt

'l-(f),a,an b-c+N:

r(a*uft:o;P

(b)-m:c:r,f3.. ,a+m-1 _-_lm m

rl'a)f # l'ta+b) ' \u.v)

*) "aFr( (c); (d);xu ,) xEFF( (e); (f);yu

l(a)l(b)"1:C;E r^ ld ;

(d);

l1)., *ro,rtl (t) ;'-' "' a,b > O; series cgt ' ov-6r the the range of integration

.- -b-l ua-L [I-u.) A.r.2.9

(l ') ; ^''"_1 o; s eries cgt. over the of i ntegrat ion (k )

a,b

G

b :(c); a+b+1.

I

I

YU

du

Lz, z'

)

".Fo( (e) ; (f) ;yuIt-"])

of integratio n

range

t Ib+ N) r( a). (a+b-c,N) t (a+b+ N) b -c,N)

..-F:G;c'[-(f):(g); " L(h)

B

a,b > o; the series of the integrand must converge over

)

:

A. 1 .2.

Ir-u]

f(u)

r(a)r(b).22C;El a, T{-aT-,r z:D;Fl a+b

- (1-u)

xaFo( (c) ; (d) ;xu

;

(g' ) ;u x, 1 vl (k' )

"n, *, K'

u*

t't,.. .l (t') ^'!

!

A. t

Io

du

r(b+N)r(a) (a+b-c,N) l(a+b+N) ( b-c,N) .--F: G+2; G'[(f) : (g) ,a+b -C+nra ; ^tH:K*2; x'l(t):(k),a+b -cra+b+n;

173

Double Hypetgeome tric Functions

A.l.2l

c ; xu

[1-u], y)

d-l

xa2(_a+o 'bici ' xuIi-u],y) .a,d>O

O

f(a+l)I(a)* o, ,^..^. (a,b ;c;x/ a>O

iffi'

r (a) r (d) r (a+d)

r

- : a,d -,O'.2;11 "'l:l:olc:a+d+1

'l_

2

4

,y)

Integrals of Euler TYPe

lA.l

L 2.15

- 'i'')

+frHP

xqr(a+L/2,b;a; xu[1-u],yu[1-u]

a > O;

"@,

"*/a1v-v/4)-b

lyl .

(a,b;c;

unless b=0, -1 ,-2 , . ,

> 0i lyl.+ unless a(b)=O,-i, c -- - c-1 (1-uJ

A. I .2. t8

A.1.2.19

u xo.(b,bi;c;xu[1-r] yu [-"] u.-1--(l-u]. d-1

- ;4'

A.t.2.21

f[2c+il

,ortu,t';-j;|,fl

,

)

b;b';-., 4 .v / 4lI

'l

A. I . Z. ZO

xY.(f+g;c,ct ;ux,uy)

;

rrlft*#r

v

rG

;r

f

*!,. ; i, v) o

r(f)r(g).I:2;0la: f, E tG;g-fo:3;ll _.^ f*s, i+s+I. ;

2(a;c,c ' ; xu [ 1-u] ,y)

:.t'

- .t,*/4,y1 c'; l

trB > U

-1

;

f,g > 0

-r; 8- 1 *Yr(f+g;f,g;ux,

r (f) I (gl -x+y -TIT.el-'

1r

,81r-ir;8-1 xVr(g+l/2ic,C'i xu[1-u] ,r'uli-ul

2

,,g

2

>

'

o

f

k.),.2.29

r(f)r(e).2:1;If f, s f+g+i. rlfrgJ-s:o;oilrr-=-r ^ f+g ZI

.,f A. 1 .2.30

o

A.1.2.31

f ;-;,

r(f)r(g)

f,grO '

,

' \!'51

f,g > o

r(f)r(g)-2:0;o[-f,g :-;-;x yl f

xu

-

1

,(a;

c1t -,.,;

i

c , c ' ; xu [

1-u]

4.i.2.32

I lf+g*l . ^. ^,4'4

L--f-''''

|

)

r(c) r(2+a-b-b' -c) r(2-b I (2+b-b'1

2a

t

2 ' 2 x/4.y/41

f,g , o

xF,(a,b,br;c;u,-u)

fo>C) ,6

I :1;

r(f)r(g).3:0;ofa,f,g :-;-; r1:rlJ-z:l;lr t f+g f*g*I..\.,\. ^.^,.

,

. s-l uf-t.-(r.-uJ" xFt(a,b,1-a;f;u,u/2)

(J*C)-

f,srO

1b'-

I

)r

(I/2)

11

x{ [ f(a/2) r(.s/2+a/2-b,)]-1

0 o; l/xl.l /yi . r l(f) r(g)

-' - *' I r)

(;i;' "'. i"' t{-i+i'F;-3:O;O,f,a,h' ; ; ' f,8 , o; l/xl*l lyl . t ;

+ft+f)*. (b,b', r, g f,s, o; l*l,lyl .

;

c

t

;

x,v)

;

Integrals of Euler TYpe

I l{0

lA.1

1

--TJ; A.1.2.69

'

,f-t A. L .2 .70

-.s-1 --

(a,b,b';c; ,*, Ii-u]Y)

*F.

6r-r1e-'

xPr(f+g,b,b';c,c'; ' ,*, [1-u]y)

,rf-1(t.,r)t-' A.I .2.71

, F,

'

(a,b ,

b' ;c,g

'

;

[1-"]yl u' '(I -u) t ' ux,

€- I

A. | .2.72

A.t .2.73

A.t .2.74

o- I

,g; "F"(f*g,b,br;f ' u*, Ii-u]y) ,rf

-

I

s1t -,.,;

"F,(f+g,b;f,g; * ,*,

[r

A.r .2.75

L.1 .2.76

-u]y)

"tl\

,s , r(f)l(e).1 :2;2, a :b,f;b',8;-. .. c; cr :^'/ -(J-el-l:l;1tf+g; f,B > o; lxl,lyl . t f

i{i|rr-*)-b(r-y)-b' I (t+9.1 f,s , o; l*1, lyl .

A.t.2.19

"F1(f,b,1-f ;g; au [ 1 -u] , 2u [ 1 -u] f,g ,

A.1.2:80

b' ; rt

+,.rry#b, f ,g , o; lxl,lyl

.

g;

.2. BI

u'-'(I-u;ts-' * (a,b;f,*,,g; Ii-u]y)

+t++*

,f- t 1t -rr; 8- r xP,(f+g,b;c,ct; * ,*, [l-"]y)

+ft+*,

rF,

^t"tl-f

f,B > Oi lx+yl . f

*21/2-3f/2-g/2

.- ,f*g g-3f-1 1,,. xrTt z ' 2 'TL' g;4uIl-u],

(b, r,

g;c,c' ; x'v)

f,g. > o; lxl+lyl <

+{+#*i

;

i

;

t

A.

1.2.

83

f) r te) I- (f)

4u [ 1-u] , yu [ ]

-

+

f

+o

r(

I*f),2f

t(f+g;t(i+2f)l(Sz . _)+ xIt+/(1-y)] --

- , o; Iyl . f,g

)

r

,. -.\-2f L"2f t--'YJ

, Ji*,2{r_f,r) 1-2f

f+o *F,(t',b,b';g;

, o; l-.1 lYi

I

(

=7Tlr

,r$,!r,!r;

rf,g - - ^.

O

f (t) r(S)

, s-1 uf-l--II-uJ"

1

i,?;l; I;:, ;,.,,,

f,s, o;l/xl*llyl .

*r

f,g , I

1+2f;4xu[1-u], 4yu[1-u])

o-1

u^ -(I-u)o ^ xF, (a,b;c,c'; * ,*, tl-u]y)

L.1.2.82

- b) | (+

'

- s-1 uf-l--(I-uJ"

(a'b ; r+g;x+v)

r(&#t, (+t1

r(r) r (g) t tlt trfisl (-*) -r ,f-16t-,r;8-1 ,, ,f *g,1-1 , . .,€r-:-7;1. g r.,r.r1r,r-#:e,rtLff3t

ayu[1-u])

13#P,1-*-y)-b f,g > 0; lx*yl . f

-f

^rr/2-f/z-g/z*b

o

,t-'rt-r1 *-' 1

r 6|1 1- z*;

o

c

(r) r (g)

r

4xu[1-u],

A.

1

f

)

4[2x-1]ult-"1)

x' v)

(l1P)

(r.s)rry

,6,

2

du

f-^

1

a, b,

2,6'

4x[1-u],4[2x-1] "u[1-u] ,rf-1(t-r)g-I

{ffirtr(b,f,;c;x)

xrFr(b',9;c';y) o; l*l,lyl . i

r (f ) r (g) r

,f-1(i-r)g-1 _, ,.f*g _ f-g*1.,."

x, y)

f(u)

1,,

^t

xF,

f-I

A. | .2 :77

f,s, o; l"l,lyl . t

(

'Fr(a,b,b'; " ux, [1-ulv) --E1. a-l u (1-uJ"

r (u)

au

r(f)r(g) -l:2;2, a :b,f ;b',g; -iIT;s)--2:o;o(c,f+g: -; - ;

I

f ,g;

fl

r(uj

I '0

f (u)

l8l

Dotrble Hypergeome tric Functions

A.l.2l

-ul) ^

-' o; f,s, l*-yl.lr-yl {-1-s-f-. 1 * r (r) r (s) rt!#l ri8:f1 -61

@

x.F, ( [f+g] / 2,b ; lf+g+\l

/ 2 ;V / a)

-

lA.l

Integrals of Euler Type

ll{?

,1 /o f(u)

f (u)

rffir,

uf- I i1-u1 8-1 A.1.2.84

€+o € o+i wE "'1tr'_ b 16 ^,-.

2 '2', 2',b'

4x I I

-u]

, ay I i -u] )

,f-1(t-rr)g-I A.1.2.85

xFt(-s,b,b';s; xu [ 1-u] , yu [1

-u])

€_l c_r u' -(1-11s'

A.i.2.86

*F

€+o

r

(t''b'

b' ;c ; xu[1-u],yu[1-u]) ,

Double Hypergeome tric Functions

A.1.2.92

*2,|\ F rf+g f.f+g+1.x1.,, 2 '2, 2 ,1-y, f,s,0; l*-yl .1l-yl

f,g , o;

l*1,

lyl .

A. I . 2.93

+

r(f)l(g)-2:t;1,f, s :b;b';x y. rf f-gl-Z:0;O\^ f+g+1. .4'41 2"

f,s,0; lxl,lyl .

A.1.2.94

A..r.2.88

*F

,(t+i,b,

b' ; c, c ' ;

xu[1-u] ,yu[1-u])

..f-1.,...g-1 u tr-uj A.1.2.89

f+o

xFr(jj,b,br;c,cr; xu [ 1-u] , yu [ 1-u])

A.L .2.90

4.1.2.91

\.-

A.1.2.q5

e t.

f,g ,

+ft+fr*,(r,b,b :.,., f , o; lxj*lyl . a ,

;;,L4)

g :b;b';x y. f(f)l(g).2:1;1,f, f(f-g)' 1:1 ;l'f+g+1 . ^. ^, .4' 4'

-f-'''' f ,E > oi lxl*lyl . a

'

A.1.2.95

A.l

-(a,a',b,b' ; f; JuxII-u],uy[1-u])

+fiHfft

3(a,

f ,0; l*l,lyl .

+

i

*l;f, ,Lo)

.2 .97

f

o

3:O;Ot^ f+g f+g+1.

-' z ' 2

'

tFr'5'L''' ^.- ^,.lIr 4'4)

4\

(l)r(e)"4:0;0,f ,g,a,b :-;-; :tf;gf 2: I :I(f+g f+g+l .^.rL^t '( -2*' 2 x/ a'Y/4)

r

.

'

^,t ; e.);i,Ya)

f,g , o; series must be cgt.

;,.,-ff;(b');

f,g ,

,rf-i A.1.2.98

y.

(a) : (b), r(f) r(g) "A:B+m;Br, (d) , :rf*ot 'C:D*m;Dt((c): f f+n- 1

tC m ,',

o-l ^

a', b,b'

(f)r(e). :(TiCf'

*f|[*', )

_

.4'4t

,/xl*t,,/vi , 2

0;

u8-1(t-r)g-1 xH, (a,b;g; *8 ; t -"1 ,1'u Ii -u]

2'2'a',b';x

f,s,0; l^l-llyl .2 .

at.

xi1[t-,,1 ,r,u[1-u])

u- ^(l-u)o xF" (a,b,br ; c, ct

xF

rFI /c l-,.a r \a,u

:a,b;

a,b;a',b';x y, - ; - :4'4) .

,f-1(t-.,J9-i

r(f)r(g)-3:1;l ,^'f ' s :b;b'; -rfT-gf z : I ; I tf+g f+g1-I_. c;c' ; ; 2' 2 "' ' xuIl-u],yuIl-u]) x/4'Y/4) f,B>Oi lxl*lyl o; l*l,lyl

xrIi-,r],yutr-ul)

r xuIl-u],yuIL-u]) f,g, o; I

.,,lyl .4 r(f)r(g). ).?.) I

-- . p-l uf-i (r-u)"

+

xF, (a,b,b';c;

f-I uf (_1-u)-

' xuII-u],yu[1-u] f,g , 0

f+o _ b f+o+l _ b ^.e xF'y'\ f ) '

f

-1FgT-'l :C;o\f+g f*g+1.

xF.(a,a',b,bt;g;

.F

du

:(f)r(c).7:2;2,

€_r n-r u' -(1-u)u ^

,f-]1t-,r;8-1 A. I .2 .87

l

f-l o-l u- '(t-tr)'-

-rr-t/2-g/2

183

lo f(u)

F(u)

du

+{}++,(r,b,b,'L*,i,I>

€+o

A.l.2l

ii-,-,1

f*g*m_ m

O;

series must be cgt.

8-l

I (f) I (g)

B; B' . (a) : (b) ; _rA: "'C:D;D'\(c):(d);

*". t, -u) n, v) [! i ] : f,8 ,0; series cgt.

1

rr , lr rl.u .

B+m+n; B' . (a) :

..,

.

*, r)

(b)

,

C: D+mr n; D' I (c; : 1d; , -ff-cl- "A: f+m- I ._B g+n- 1. 16, r . I m,., m 'n,., n ,(u /' f+g+m+n_I rdrl f*g '\e " m+n"" m+n mn -m+n" m"n"x/ [tn+nl" ,y)

Integrals of Euler Type

I84

i'r(r)

f (u)

'0

(t-u1s-r B;

Br

D;

o'

[(aJ : (b) ;

fic)

lA.!

: (d)

t

*A'FB,

;

[[i:

]

;,.'

A.t .3.2 '

''o[r)

Fu

f

[r)

be

tu'

,.rtf-r]

I

r)r.

Lro( ')),

,rf-1il-,r1 s-1

B; B'[-(a) : C: D; D' l_(c) :

(b) (d)

(b') : m

fi]

xrF

;

A.1.J.J

;

r{a" -N;a-g-N+1;u

* F1(f*g'a,b,b';f+g ux, uy]

ia'i,*u 'Y"J I

,f-'

f,gro;

cgt

-

x

1,

-r;

8-

(a , -N; a-g+N- 1;u ,F ,

, Fr(c^,f+g-a,b; t+g ,d;ux,yJ A.1.3 Euler Integrals involving Multiple Hypergeometric Functions

,rf'1(1-r)g-t x

A.1.3.5 ,l '0 I

t-,

( 1-u)

,Fo,

l-(,' L(n'

A.1.3.r

f[u)

du

A.1.3.6

]:,.,"] );

o(r)' u (r) lrr(')t, x

f,g,

.il ,-]

o; all seri e.s must be convergent over the range of in tegration.

rF

r(a,

-N ; a - g+].{- I ;v

x P, (f+g-a,f+g+|,,1; * dr6t;ux,uy)

,rf-11t-r1 s-r xrF, (a, -N; a-g-N+1;u)

g-1

[(r(')

xF

o'

u(1- ,)]

uf-l [--1-u]. :s-1 ,rA: '

185

,f-1 (t-rr)g-l

r(f'lr(g). A :B+m;B'*ml-(a) I -fTfT[lc+nr; D ; D' Lf.l, i

> o;

r ies nust

/l rr'l

f(u)

]:r,',rrr-rtt

o

Multiple Hypergeometric Functions

a..,

I I

A.1.31

(g,N) (f+g-a,N) (g-a,N) (f+g,N)

.F, (f+B-a+N,b, b' ; f*grN; x,y) f ,9, O; lx!,ly . ) (g,N) (f+g-a,N;.

(s;ffiT.s;Fi,

,*1

'*2'.'*r.)

N' b

f+g-N,d;x,y)

f,s, o; lxl*lyl .1 [e,N) (f*g_erD (S-a,N) (f+g,N) ,F4 (f*g-a*N, f+g; d,d' ;x,r') f,s > o;l/xl.llyl . t (g,N) (f+g-a,N)

[g*a,N) (f+g,N) (

' tft) lb,f +g-a,cr, . , f18 ,dr, . ,dr: ", i

,^ t c' f+e-ar

r'l

"Fi^' (b, f+g-a+n,c2, .,cr; frg+N,d . 2, . ,dr; xr, . ,xr)

f,g, o; l*rl*..*lx.l .1

;

Integrals of Euler Type

I U(r

I

f (u)

Io

,f-1(i-,.,)g-1 x

f(u)

a-g-N+ I ;u) rF r(.a,-N ;

(g-a

JXf*s,Nl'

B

b2, . ,br ,f *Elc2, . ,C-id;xr, . ,x.,

r'1.1

,*1'*2,.,*r)

A.r.3.8

I I

(g,N) (f+g-a,N).(r),

rF, (a', -N; a - g-N+ 1 ; u)

"

*

rF

,(a,

-N; a-g-N+

..ru-111-,'.,1

ZFi

1;

u)

(

H:K';.;K'-'L(r1.)

:

(k');.;(t('));

',*{

,f-i11-u18-l (r*e,u

' ' ;c(.)l-1r;, H:Kr;- ; x(.)L(r.,) ,

ffiFi-'(r+e-a*N,

.

)

2, -,brl

c;uxl,12,. '*r)

;(k('));

€ €r u (1-uJ

t

f,s

,

cgt.

r(f+1)r(f)

i(2f+1) -2 r" h "^(r)

*1 h

r'"'4

I(f+1)r(f)

Ir) r {2f+l) *0.\-'(bt,.,b.;f f r) ,c)'' (t1,.,br, 1*I/ -,, . " x,uII-u],.,x-.uIi-u]) 4 j--r;

A.1.3.14

r(a) (a+b*c,N) Y (a+f+ N) (b.c,N)

I(b+N)

A.1.3.15

f) : (g') , YF-F:G'+2 ;c",.,c(t)[{ H: K!+2 ;K",.,Ktr)L( h);(ki), - [r]a+b-c +N,a ; (g,,); .;tc.-l; a+b-c , a+b+N; (k") ' . ; (k(') );

1

f ,E

r(f)r(g)^(r),. ,bz,.,br;c; ,, jffi")''(f xI'''xr) f,g ,

,.,f

A.i.3.16

-

.vj')

i

1r

I -,r; 8-

(*; r,d2,.,d T;

uxl'x2"'x r)

*r , . ,*il series must be cgt.

>

€_1 a_r u' -(1-u)" ,oj') {ur, . ,b.; r;

uxr ' ' 'uxr)

u-(1-u)

4.I.3.17

o

A. 1 .3. 18

fr\ "Y)' ' (a;{+L,d), . ,d"i xru [1-u] ,x2, . ,xr)

,f-l1l-,r1 8-i ^n5')

G*g;cI,.,cr; uxr, . ,ux.)

't

;

f>0

f+g;dl, . ,dr;x1,.,x.)

f,s, o;l/Iri...*l/xrl .

;

(g(')

I ux,,. I 'u*rl € f_r u | 1-u] 'a\') (f+I/2,b2,. ,br; c ; x1u[1-"],xr, .,x.,^)

r' l.t

o;

;

f>0

oF:Gt;.,c(')[f O '

,oj')

r,at.

F' ''

(k')

r+8- a+N'

b-t

(g');.;(g('));

A.l.3.n

A. 1.3. 12

(g,N) (f +g-a,N),-(r)

(., -N;c-b-N+l;u)

"*r*r,

(1- u)- b-1

)

c],.,c-' f+g+N;xr,.,xr)

f,g ,

* p(r) (f+g-a,f+g+N; c1r,.,d.;uxl,.,uxr)

A.i.3.10

{-;;.rj11..sjiFi''

a,

"^Fr(c'-N ;c-b:i'l+1;u)

oS') (f+g-a,cr , . ,cr;

. f+g;uxr, . ,ux.) .,f-1(t-,r)g-1 A.1.3.9

^

(o I l

uf-1 (t-r)8-l x

u*

187

f(ul

I] '0

f (u)

du

f,g ,

f+g+N, c2,.,cr;d;

Multiple Hypergeome tric Functions

A.1.31

(e,N) t f+g_a,N)-(r) gf*g_r*N,

* p(r) (f+g a,br,.br,

A.l .3 .7

lA.l

TPr.rr *i')

1u;r*g ,d2,

',d

xl,.,x2)

f,g>O

++t#{f\ [') {u;*1,a,,,a xr/4, xr, . ,x.)

f>o r (f r (g)

(r)

) ... ^ -rf?;[f+z'tr;ci'''c ,,,

f,g ,

O

l A.l.3l

Integrals of Euler TYPe

.I jo f(u)

(t-,r) f -r "y)" (f* L/2;dr,. ,d,.i ,-,f

f(f+I)l(f),(.) '2 l(2f+l)

rl

.*r'-*)

f,g ,

[1-u1xU*r,., Ii-ulx.)

h.7 .3.22

h.1

.3

.23

A. r .3 .24

*r[t) tr*e ,a2,. ,a-; bl,. ,br;c; ux'xr,.,xr) ,f-t6t-rr;e-' ,n[') (u,u; f ,d2, . ,dr; ,.,f

A.1.3.26

\:

-

i

1r -r.,1

8- I

*r[') {,, f+g,br, ' ,b c;uxl,x2,.,*r)

A.1.3.28

b1 , .

f,g

,b.;

o;

(',0

3.29

2

L.

,b2,. ,br;

1

u2,.

1

lr .r.

'r[r) {r*e,u1,.,br;

dr,.,d.;uxr,.,uxr)

A.t.3.32

A.1.3.53

o;l*11

. a,!*rl,.,lxrl.t

(,,r

I

,r,, ,o,,Yf

.,dri x r/ 4,xr, .,*r)

f,s, o;)xr/+l*.[xrl*.*lxrl . r (e) .

g,b

f

,s

,

z,

(r) ,.

1

^

.,br; c;xrl-1, xr, .,xr)

o; l*r

|.

+fi*fi!-[')

a, l*r

l,.,l*rl't

r',0 ;*1,a,, ',a

xr/4,xr,.,*r) f , 0; llxr/zl+l'ixrl*.*l/x-

+ft+*l')

(r,br, ',b,;d, ,' ,d,i

xr'''xr) , f,g 0; l*rl*.*lxrl .

1

,.,f-11r-r1 8-t

n[') r.r,.,ar,br,.,b f;ux1 i.,uxr) -- - s-l uf-L (i-ul"

f,s,

d2,

,^,

-- - s-l uf-1 (I-uJ"

'u

xr' ' 'xr)

+HS\l')

::-f)

'"2 xru[1-u] ,xr,.,x..) ut-- -f-I

A.1.5.31

lxr/al*l*rl*.*lx-l .

0;

'xr) g-

^n[t)t,,b;r+1,dr,', dri*1rIf-u],xr, . ,*r)

r

f ,

1

:

[1-uJ

-e,o2,,d.i xr,.,x.)

f,g, o;l*11,.,1*rl .

-r)

,rl') ,*, f*g+I

;'1

+i++*'S') {u'r'b,'

r,,f

t (,

,t-

t

l/xrl*.*l/x.l .

"rf')

o-

xr ' '

xl , . , xr)

, o;lrrl,.,l*.1 .

+i++P'[') f,g ,

c;

I

g,d2,.,dri*lrIi-"],

r(f) r(g).(r)' tr'a2'' .. ^ 'ar' r Cn eI1 B

,r[')

{u, r*!,br,, ,br; c;ux, [1-u],xr,.,xr)

f

LGUIg) r-*r) -ur['-r) (^,b2, .,b.i I (f+g)

-11t-,r;8-1

"1 '*2' ' '*r) A.1.3.25

0

,r.ft) ir, f+g,br,. ,b r' d2,. ,dr;xr/ [l-*t] , . ,x./[1-xr1) f,g,0;l*, 1*.*lxrl . 1 f,d2,.,dr;ux,,x,, '*tJ . s-1 r(f)r(g),,(.),^ . * uf-1--(r-uJ" frl' (a t, ffi',r''ta,t,b:,',b, , E,b Z, . , h.i dl,',dr;x,,.'xr) ^Fidl,,,d.tr*f2,.,*.) f,s, o;lxrl*.*lxrl . 1 ,..,f

u-- -(I-u)o

r/ 4)

,p(t-l; (a,b2,. ,b.rd2, . ,dri axr/ la-*rl, .,4x, / [4-x1 l)

t,dZ,.,dr;xru[1-u], *2' . '*")

(Bi,p*1, . ,drix1*1, . ,x.)

(.1-x

f(2f+1)

'r[') ir, t*l,ar,.,t r;

,,,

iJXl'..,uxk,

A.1.3.21

.3 .27

I (f) r (e) (r) ,.. tffirl *)'' (f idl, , . ,do;xr, . ,*r)

*v!t) {r*s;dr,.,dr;

1] I o-l u' '(I-u)" -

A.l

f>o

s

i(f+1)r(f)

u^(1-u)

A

xr/4,. ,xr/4)

*ri1r-u1,. ,Xru[1-"]) u'-'11-u1

/l tr,1 o,

du

r.. A \"el'''v

t89

Multiple Hypergeometric Functions

*r[t) {r*e,b;dr,.,dr; ux.,.,uxr) f,g , O

f+g;xl,.,*r) f

,s,

0

i

A.2.1

f

:*;xu m

/t f

4.2. 1. 18

A.;t.l.lt

,t-trr, riil.*"zt

z'

"-'r0,. \:r

I

F

/f

r

ruj i- -o'x=,2) A S B+1 terminates.

f,B > O; lrl

on

unless AFB

--. t;

termrnates.

ff,j i,,.)

, "r .;,[r.l [iir

f :(a):-;.x2,_z) _f tr:B;o(f+I:(b),-, L /t,..1:A;O, fr0;A.BunlessOF,

terminates.

' 'an'i-r' ,t .oTr:llir,.ia,' r ''a -

ibr,.,bm, -f,

urr*1'''uO

b-*1'.'bq)

/f ,F ,(a,f;b;xz) f> 0 lzl 5 1; I*l z'

,onu

,F

.

f 2r(a,b;i + I;xz")

f , o; lrl : t; lxl .

t I

f z'/f ,rr(a,b;f+I;xz)

f > o; lrl : t; l*l .

I

?,I.ri.,^,

S BorifoF,

o

f>o

,t-'rrr(u':*r;-r.,)

'ahs(i5j.uxJ

l*l.lifA=B+1.

,

f :(a)'t-"' z-f,r/t Fr7:A;1, r,e;o(r-r,

,f-1 "-u A.2.I.17

o*ro u*, , [i]

terminates

re.'striction xifAo t

-t

A : AFB

2-t rFr(i* 1;b;rr )i z-/f f LG/2;b;xr21 f>o

+I

,t / r,r rr7,,l,.f',tr;"r' I

,f

f>0

f-I _ -F. l-f ll-m

,t''ro, i";l;*,r2) ,,r-,o.r{lf,];.J,

I

A.2.L9

A.2.1 .14

,f/f ,rr(a;f+l;121 f>0

,Fr(f+1;b;ux) ,

,r

f , o; lrl -. t; lxl .

zf /f. _ explxz2.) f>o

f-l _ .f tFrtT* €_l

,t-'ro, {";b:,r*)

a,

,f /i ,r r(-;f/2+r;xz2) f>o

'-'rtr(f+1;f;ux)

A.2.1.5

/r ,Fzrl:?;ii.,l flro;lrl:t;l*1.1

4.2.t.t3

,f/f orr(-;ft1;zx) f>o

(-;f;ux) J

A.2.r.2

/i rt"1 a,

f (u)

A.2.1 Definite Integrals involving Single Hypergecm3tric Functions

193

I

(r,l

Definite Integals

lA.2

Double Hypergeome tric Functions

A.2.21

/i rr"; a,

4,2.2 Definite lntegrals involving Double Hypergeometric Functions €_'r

L.2.2.1

,t-t*r(f

,r/r

* 1,b;c; xum,yJ

f

u- -Y2(i*1;.,"'; +-_l

4.2 .2 .3

F

xumm. ,yu

fl

4.2.2.5

A.2.2.6

(r,* * l,b;.

xum.,YJ

)

fIrS-,;r,b, mm. c;xu ,yu

A.2.2.8

A.2.2.9

,'-

I

'

A.2.2.tO

F

F1(a,b,b';:; xumm,yu J

,t-t t, (a,l + 1 ,b; fm -,c;xu ,yJ .-l



rr(a,fr+ 1,bi

-m c,c';xu

,y_)

r

r(f;.,.,

A.2.2.14

/f ^tr;i* f>o

f_t f (:+ u- - F-l-m 1.b.b': fmm-.yu -:xu m-

+_

A.2.2.),3

ff

,'

t

a,

r(b,b' ;f * 1 ;*r',yrm) ;xrm,1,zm;

(* * 1,b,b' -mm-,yu c,c';xu Fz

;

v

1,c;xzm,y)

,f /f ,r(r,*,u; c;xzm,y)

f >o

(

t

r

-r*r) -o'

l*=*l,lyr*l .

ff

z'/f

F

l*r*l,lyr'l r(a,b,b' ;i*l;*r',yr,)

f, o;

o

.

lyl*lxzml

r

f r{},u,u, ; c,c, ;*rt",yr^1 f>o l*r*l*lyr*l . i .'

/

F

=f

f>o ue-l

u x,u

yJ

-u

l*r'l,lyr*l

:(a)

> o; lyl*lxzml ++mz' / f F ;xz"',y) r(r,r,b;c,c'

.

t

f'o;

.

lyl*lxzml

t

:-: f ;(b);(b')

=f-(s),-: t -....r^r. 7r rLl.-.r-r..,1 i (d); (d');

;

f , O; series must be cgt.

A.2.2.18 ux, y)

f-1 ue 4..2.2.t9

B; B'



-u (a) : (b)

;

(b');

z^-(3), f ::(a):-:- ; (b) ; (b') ; F- tf*t: : (c) : -: - ; (d) ; (d') ;

^.A: ' C: D; D' r'(cJ : (d) ; (d'); -LLrlLr-L)

t

;.

,rf-l(t-r)g-l

-'rr, ('io U -xzm; -1;-r_l L-xz

f

]'/1xzm; l*1,/1yzn;

L.2.2.t7

. .

,t / r

>

u'-' tr(|. t,u;c,c'; : ='/f o(|-,u;.,c';xzm,y:m) m m.

ux,y)

f >o

f

Fz(a,b,b';t *f,c;xzm,y)

/f Fr{^,^',b,b',f .r,*r*,y{1 f> o i*r'l,lyr*l.i ,f-i Fr(a,u;I,.; | -f tf rrtr,o,f,. r,c;xzm,y) *r',y) i/1x:)l*1,/yl .r if ,o uf-1 Fu{r,ar;b,bt; fmm-;xu ,yu

, .)

/f

F

A.2.2.L6

lr'*l,lyl .

f >o ,f 1 f 1r - r^*1-o

.f

J

f>o

J

f u^ tm xY2(a;;,c;xu,YJ

.'-'

o

o

,f / f

..1

4.2.2.4

A.2.2.t2

J

,r / f

,lr(u,u';i;*r*,vr*)

u- ^ F2(a,b,br; fm -rc;xu ryJ -1

or($,u;c;xzm,y1

f>o

f -t

A.2.2.2

A.2 .2 . 11

/f rr"l a"

f (u]

195

-\

ux, uy)

f>o

series must be cgt.

Definite Integrals

196

lA.2

+

,t-'1t-r184.2.2.20

...A'

^'c

'

B; B' , D; D' t

(a) : (b) ; rt-.'); (c) : (d) ; (d');

f : :(a) :-:-;(b) ?-(3), \f+l::(c):-:-;(d);(d');;(b') f ' I'

uf-

;

A.2.3.6

-o' "'.*zryzrz) t

ux, uy)

f,g,

/f

/f rr"1 a"

f (u)

lr l -. t; series

o

.. ..,A(r) (r,** 1,br,.,b; dl'

'

,dr;umx,

,xr,. ,x)

must be cgt.

,ri') r,,* * , ,0r,. ,u; A.2.3 Defi nite Integrals involving Multiple Hypergeometric Functions

A.2.3.7

/o'r{r) 4.2.3.1

,.

jl, ,f *

L

,b2, .

,br;

c

um, xI'x2'.'*.J

*oj') (b1' . 'o.rft x-...u umm xrI'

u

'*j') ,* + l;cr, . ,cri umm x1'.'u

ufA.,t.tl,5

xrJ

I

*r[") {r,ur, . ,or,*, rlr,.,dr;umxl2,.,xr)

#1"

A.2.3.8

?*t', (br, . ,b,;! * 1; mm . ' z"'.xt, ,2"'x_) f>o

A..2.3.9

*1,*2,. ,xr)

f>o

A.2.3.10

{f; ., , . ,.,; mm , *lr.,Z Xr)

lrl € ,Fi''(;+i,h,,.,b.; mm c,,.,c -;u x,,.,u"'rj r' - T' t' r

-(rl 't;-f * l 't-; 'a2, ' ,ar, ol,.,br;c;u m xl,x?, ....x -r- )

itl"

.r[t) {"r,.,a.,br, -,b.i fmm,i' *I, . ,u 'aJ ,-,f

-

"rl"

f>o t L--

(a,br,.,br;!* dr, .,dr;

au

,",*,br, -,b ;dr ,' ,d,', z"'xl,x2'.'*.)

f, o;lr'*rl*l*r1..*lxrl ' .rlt-')

I

(u,b2, . ,br;dr, . ,d

xr/ il-znxrl

t./

,.,

lz**rl *l*rl*.nlx-1

m ,IIl.. Z X-,...r/'^-,) * f ' O; l,' rt"x ,t

'

'

1t-znx17

)

1

r'*r,*r, .,*r)

f > o;lr'*rl*lxrl*.*lxrl .

c* + 1,br. . .b

I

.

*

m :"'x.

l .

.1

m. , *\,*2'.,xr.J

f , oilz'xrl,1""I,.,l* +1"

(a, ,.

,a,,':,,.

,u,;{'

r;

,^*r, ., z'*.)

f, o;lr**,1,.,Iz*xrl .

1

c;umm xl , . ,u x.J

1,

rl*

r

,f -1r1 ,f ^ T-s tm,'-2'.'"r

f

.m-.

*5"

rt,1

u

xy

x2r.. rxr)

.vj') t.;f,. z,.,cr; n'*1'*2'''*rJ

A.2 .3 .4

n

u

)

L.2.3.3

\I, f>o

f>o

u

N.2 .3.2

;

f*:

t,m i,d2,',dr;u x2'''xt)

a,

z' -(r)'(r,bz'.,br;c;z ,f

t9'l

Multiple Hypergeometric Functions

A.2.3\

*5"(*,0', mI

,b-;c;

, *l,.,.

f

tr

rlTlrln > O;lz"'.r,

rrJ

,.,1r"'x.l

I

lA.2

Definite Integrds

198

If tt"l

f(v)

ri f{u} du

f(u)

o-

f I u

,. r,!', {",* * 1,t 2, . ,b;i A.2.3.12 m--\ C;U Xl r)(2t . r^r) u

(",*,b2,.,b,;c;

*,5"

f

,r**r,*r,.,*r) , o;lrmxrl,l*r1,.,1*rl . t

*r[') t",u;f ,c2,. ,cr;

(a,u

+1"

iro

;f * !,cz; ' ,cr;

m

z"'xL,x2'.

, m. *1,*2,.,xrJ

L.2.3.t3

A.2.4.3

' c-1

L.2 .4.5

"r[t) {r,ur, ,0.,}, umm 41' . 'u xrj f>o

l*;"(a,br,.,br;|*

r;

zmm, xl, - ,z xr)

N.2 .4 .7

lr**, l,.,lr**rl .

uf-

xOFr(-;f;av)

A.2.4.1

'oFt(-;g;b[t-v]) g- I vf- I (t_vj

A,2.4,2

"

11 1

(

f*B

;c

; [x-Y] v+Yt)

f,8>O

1

o -1 o

r(f)I(s) TfT;sl-

:f

*f+g-1

xrF, (f ;d;xt) rF, (s ;d I ;Yt)

vl t

o

1

xlFt(f+g;c+d; [a+b]t) f ,g > O .

'2 ( f + g; d,d ;xv

(t -v) J vz ( f + g; f,g XV, _1

i(f)r(g)

_L

ylt vl

-f+g-1 ^(x+Y)t

r I r+9.1

f ,g

,

o

t

.nrr(

\t ) ,7,

[;] : ""2)

(e)

[]l;u [.-,]2)

I

2l

f,g , 0; series cgt.

,t-I 1t-r)d-1

I

_l

r(c)I(d) *c+d-I f Ic+d.]

.rr.r[[]:,,r

/f rt'l B- I 1. _v1

[t -vl )

o

o

r(;d)-'

(e;d;b

(t -v)

?

I (c) I (d) -c+d-

n

FFc(

A.2.4 Definite lntegrals of Convolution Type

1

f,g

)

,t-1 1"-r)d-1

f .

A.2.3.ls

A.2 .4 .6

r (f+g)

11_v)d-

ty

;

XrJ

,rf-1

,Fn

a"

r(f)r(g) t f+g-1.(a+b)t

,Fr(f;c;av) f

xr[r)t.,f+1;cr,.,c.i , mln *1, . ,u f>o

x

'*r)

l*llxrl*.*l/x | < t

l,/(zmxl)

A.2.4.4

,rf-1

A.2.3.L4

(t -v) g;f,;av) "tF1( *rFl( f;g:b[t -vl

x

f-1

199

Convolution Intcgrals

A.2.41

A.2.4.8

a'

f,.g > o

r(f) r(e) *f+g-1

-Tr*EJ-' f,8>0

-f+g-1

r(f

'g;

series cgt. r-(T;E

A.2.4.9

xo

t

at,bt)

r(f) r(g)

"oF1(-;f*g; [a+b]t)

r(f)I(c) -rIT-sI-

.rtr([]l 1u1t-u11

-O:F+I;H+1. - :c,(f);d,(h); (g); (j)

"l,G;J\c+d:

-e

- *c+d-1

- A :B+t;Br+1, (a) :(b),f; "tc*l, D ; D' \(c),f+g: (d) ; (b'),9;-.\ulfliiixt'rt)

'

c

;

xt Yt) '

series cgt.

F-epeated Inlcgrals

200

/f lIt

-d-I

!rl (cJ;",r :

Ii]

;o

rr"1 a,

,t-'rr(b,b,;

- : :-:d:-;(f),c;(j); . \^. .c+o::_:_:_; r. . ""(5), (g) ;(k);

'..,,,

fl]:,.,bt,qt) (pJ ;

r [f ] :o r.-,r

,d , O; series cgt.

- - e-i. vd-I [t-v.)

v

A.3.1.4 A.3.1.-s

2.4.

r

r

xz oyz)

,

c

t-a-T*,

(b, b'

z +r- 1

,-

ft-v]

A.3. 1 .7

,-

(f+r,b,b';c; . xz ryz)

f+r-

;

,e

=-

>

A.3.1.10

A.3.1.11

r {? .ft . . l'r7) (dz) f+r- I' z-'

ITJI-

.

A+IFB*r

r (.(a), iuj ,r-r,*')

ffi'-= I

fu.r*,

(a,b,br; f+r;

,-

"

xz,y)

lxzl.lyl .

,c;

f_t ,- -Fr(a,f+1,6. - c, c' ;xz ,y)

12

,'-'rr(f+r,b,b,; - c , ct ;xz ,yz)

+r-1

r(a,b,b, ; f+r, c;xz,y) lxzl*lyl . i f+rI Z^ G,rI+z (a' f ' b ;c'c' ;xz,Y)

d

'fr;;f* r(r,b,

A.3. I.13

_t

z'-F.(a,at,b,br; " f ;xz ,yz)

f+r-

I

?CO*a

;xz,Y)

I

b' ; c, c ' ;xz ,vz)

jxzl*fyzl f

(';o; ,{..)

t

lxzl*lyl .

A.3.1.

xz,yz'1

t

-arF,

.

t

(a,a',b,b' ; f+r; xz,yz)

l*rl,Ir'zl .

f+r-I

k,-;f*r(r,b;c

f+r- I

ffiCr-x:)

xz,y)

F-t'Fr(l,b,h,;f

A.3.1 Repeated Integrals involving Single and Double Hypergeometric Functions

series cgt.

,Y)

ot

l*rl,lyzl .

A.3 REPEATED INTEGRAIS

,r-Iorr{ti] i.,,

z)

(r,b,b,;c;xz,yz)

f_1 -Fr(a,f+1,5.9,..

(i),e; (i'); at,bt,c) (p) ; (p') ; serie: cgt.

;

;

xz ryz)

r (d) I (e) 1 -f G-e) -d+e-:(1r):- (f),d; ,F(3)r-:: ' \_::d+e: (k)._ ;; (c);

.r.FG([;]:"") ;

Y

l*,1,1].zl < l ,' 'Ft (a,b,b';f

bIt-v],c)

1

H,

)

*rH:J;J'.(h) : (j) ; (j') "' K: P; P' : (k) : (p) (p')

r ; xz,

'f-a--;f*, (a,f ,b;c;xz

xz,y)

i*,i,lvl

vd-I tt- -vJ- e-

!1

;f+

ailD+z(f ;c,ct;xz,Yz)

fI

4.2 .4 . t2

+r- I

-lFr{r,f+r,b;c;

€_1-FI

;

It-r'l

z

(dz)r

r(alf ,c;xz ,y)

4.3.1.6

_rH:J;J' r(h) : (j) ; U') ; "'K:P;P" (k) : (p); (p') b

f ;xz,yz)

f _'l ,- -Y/ f+r;c,cr;

'o'.r [f] ;""r A-

li .trl . .l'r(r)

t (z)

rIc]r(d).c+d-1 -T-6ar

-vJ ;

4.2.4.10

Single and Double Hypcrgeometric Functions

lA.3

r

r (z)

A.3.i.14

A.3. i. i5

11

,f-Ir.If*r,a,b,b' " c;xz ,)')

l.

,'-',

f+r-I

fo;rf

f(

1611'

t,a,b,b' ;c"xz af

'Y)

L.3.2.3

o(a,

b;

f , c',xz,v)

f+r- I

?t;il*o

* 1f*r,b;.,. xz ,yz)

z

f-1 (a) : (b) (b')

; *,A: B; B' r "' C: D; D' ' (c) : (d) ; (d')

xz,y)

t+r- I

k.3.2.4

t -1 ,vls) (u;f,c?,.,cs;

b

c'

,vjt) (r*r; Y

ct

z)

;xz'Y tr-,r:*o ' ; l/1xz)l*l/1yz;l . r zt+T-l -A:B+I;B'.(a): (b), f If,-Tc : D+I ; D' \ 1c) : (d), f+r !! (d ll ;' , rr ') '*,

B; B'

A.3.1.18

"_ .A: ' C: D; D'

(a) : (b) ; (b') '(c) : (d) ; (d') ,.

\ : D;D'

(r-j-c*t

: (bl f+g : (d)

f

(c),

A.3.2.s

Y

z,x-,.,x

)

iala

4.3.2.7

f-1

f s'l xF'-'fo 415 c^ L tw2t.

both series cgt.

h

A.3.2.t

A.3.2.2

f-r *o5u) (bi,.,br;f; xrz, . ,xrz)

tT;f*)-'

IT,.)-(

1-x.z

)

*a(s-i ) (",b2, . ,br! c2,. ,cs; xr/ ll-xrzl,.,xr/ [1-x, z])

A.3.2.8

tLrt

'PIt) (r*r,u1,.,br; ,XSz)

*r[t) {r*r,, 2,. ,ar, Lu1

(r ,bz' ' 'b=;c;

L .^. r ' r u= r L r

*r''*2' ' ' *r)

*lr,*2,.,*=) z

4.3.2.9

1

(a,br,' ,b.; r+n, c 2, ' ,c ri*1',*2, ' ,*=)

,f -l

*ol-'(f+r,b^,.,b^;c; l'z> xlz'x2' . '*r)

I

ft;,+ttl

h

xIz 'x2' . 'xs) f-1

A.3.2 Repeated Integrals involving Multiple Hypergeometric Functions

,.".,,

' 'c ,i ,x s)

.

zr+r- I ,,\s) rc.^ IErl-z'iricI'''tri *Lrr. rxsz)

f+r-

CI,',tri*1','

.f+r-1

lsl' (a ;1+r 'c2' x lz ,x2,

lx.zl*lx^l*.*ly. I o

r(a+u)r(f-u) xY^(a+u;c,c';x,y)

.7

ux, [1-u1,.,rXs It-"]) A.5.

@2

(f-u)

t

j?:t'r',

2"'

l(a+u)r(f-u) l---eI ,Fr(a+u,b,b';c;x,y) )a+t

(a+r; c,ct

;x/z,v/2)

a+f>o (a+f ,b,b':c;x/2,y/2) I, -

a+f>C

l(b+u)r(f-u) I-(b+f)/c (a, krr b+t, ,b' ;c ; x,Y) r xFr(a,b+u,b';c;x,y) F;F-+ r

b+f>O

au

Y-i(a. +cl f''-I ''

F rat+c,6r_,.,ap',*12) 2'r*c R'nt ' 1u) ;

,, r (b+u) f (b' -u) .\.5.1.10' rF,(.r,b+u,b,-u;c; ^Ir' x,y)

*-rc$Jrr, "

(a,b+b '

;c;fx+vl/2) bib, > o

al*c'o

A.5. 1.2

I (ar+u) I (ar-u) ,oF, (a1+u, a2-u,

as'''aA;x)

(b)

t

I (b+u) I (b' -u)

xrFr(b+u;c;x) ,rF, (b'-u; c' ;y)

ar+a, ar+ar+l (--;-*' F 2 2al*a2 A'B\ a3,',46i*1 (b) ; ar+a, > o

t(a, +a^)

{fr$)*rro+bt 2

b+br >

;c,c ' ;x/2 ,Y/2)

(f-u) (a*f ,b,b, ; c,c, ;x/2,y/ i IfP*, xFr(a+u,b,bt;c,ct; i2 a+f>O I (a+u) I

A.5.1.

12

2)

x,y)

r (b+u) f (br

A.5.1.13

-u)

O

A.5.1.14

(f-u) 'Fr(a*u,at ,l:1-, ,.t I (a+u) I

'I

r

fb+b'l F, -

(a , b+b' ; c, c' ;x/ 2 ,y / 2) - b*b'ro

2"'"

fIa+f)'2a*t

_-j--;-1.-

5

(a+f ,at ,b,bt ;c;x/2,y)

a+f>O

{s.lI

lntegrals with respect to Paramcters

218

Y+

1-

/ i-

f (u)

Inte*als of Barnes

TYPe

Y+

Y-

I (a+u'| I (b'uJ

A.5. i .15

xFa(a+u,at,btu,bJ;c

!

r (br+u) r

ffi.,*'";=r,#'icix'Y)

x, yJ.

A-5.1-22

:i,

I (b+u) I (b t -u)

A.5.1.16

A.5.1..17

H#+

xF-(a,at ,b+u,br-u;c *rY) ' I (a+u).I (f u) xio (a+u , b crct;xrY)

LG*D+,f

(b-u) xF, (a+u,b-u;c,cl '+ x,y)

l'(a+bl x, y.) j;lr----t4( 2 ,--T-; ;.c , ci ;

I

t (a+u) I

A.5.1.18

A.5.1.19

I (ar+u) r

(f-u)

xFA:B; B'

Iar+u,ar,

|

.,rAt

C:D;D'L (c)

B

^+f>O .a+b a+b* I

2^'r*f c:D;D'f

(c)

x FA

A.5.1.21

:

lfit.,fk++---qt-

r (ar+u) .c,.5. r.25

,

jji-

f: r*^, -A : B ; B,

a

B; B ' ( a) : b, +u , l-

c:o;D'l_(c): br, . ,bu; (b') , *,J

B;B'f(a) r+r,br,., bsi Ilan: c:o;o'l(c): (d)

J

,br;

A-5.1.26

${!*1'l

(a,

b,+r,br. .,b,

;

.1,.,"gr*,*r,.,*,

r(ar]br)a{s)

I(b,-u)

(ur*,, ,a2,. oar,

241*Dr 'D

grr*br,d.,.,a_, L >

z

br-u,br,.,bs;c;.

. *1'''xr)

,or[l) 1a*u,b; cr,

.

ncs i

I (a+u) I (b-u)

"rlt)

("*,r,u-u;cr, - ,c

'xr) (f-u) I (br+u) I

*r[t) (",ur*u,br,.,b. c;xl,.,*r)

r.

f(a+f ,a+f a+t

{..! 2a+b

a+b

xr ' '

A.5. 1 .28

.,.,i

b,*f'O

.*t':'f s) A.5 ; | .2?

(a+r;cr,

xr/2,.,xr/2)

' ' 'xr)

I (a+u) r (f-u)

2br*t

(b') ; I ( 4'1rx/2'v) br+f > O

.

>

a+f>O

1,.o

o' tb

'

u+ar+L

,,D;D'L-7-r";7al*az'

(f-u)

(d) ; (d');

x1

:

i

rj"

cl ,. ,cs;

*rIt)

*ry] r (br+u) r

(c)

br+f

'

' xr'''xr)

,rft) {",ur*u,br,

ffi:'),o, ff?ifl to,.

*njt) {.*r,c1,.,cri

,a Z! .,a Al

aa,.,ao: (b) ; (bt) ;*,yl : (d); (d'); l

e-3,.,agi (b);(Ui) ! (d) ; (d')

*olt) (ur*r,br, . ,b.; . c;xl,.,xs)

r(br+u)r(f-u) A- 5. 1.24

I (aI*f).A: B;B' l-ar+f

r (a, +ar)

,'[i, +17,dz=l , D'L

A.5 .1 .23

,b;c,c, ;x/2,y/2)

:

i (ar+u) I (ar-u) xt-A:

u+f

(f-u)

I (a+u) r (f-u)

a+b>O

lll;8:l;,.'l A.5.1.20

(b+bt,a, a' icix/2,v/2) b+br > O

1-

/. r1"i au Y-16

f (u)

r1u1 au

r tb-

-tr 2-L

>.

O

+f)

J

.?-+;s (a, br+f , br, x,/2,x-,.,X^) LZ5

>0 b-+f L

., bs ; c ;

Integrals with respect to Parameters

220

lA.s

Infinite Integrals along the Real Axis

A.5.21

A.5.2 Infinite Integrals along the Real Axis

r

J

,"_

Ir(br+u ) r(f-u) T EI A' BI.

lor*

I

_1-

(aJ

;*l

u'b2''

'or'J

(a) b

2-u'

bs'''bs

lr(c+u)r(f-u)l-1 *Q2(b,bt;c+u;x,1')

A.5.2.s

I(f-u) ]'1 II "Ft(a,b,br;c''u;x,Y) (c+u)

Il(c*u) r(f-u) ]-l I I 'F, (a,b,b ;c+u,c ;x,Y

II (c+u) I (c'-uJ ] ,Fr(a,b,br;c+u,cl x

A.5.2.8

A.5.2.

II (c+u) r (f-u) ] *Fr(a,b;c*u,c';x,y)

10

lr(cr +u)r(f-u)l:1 ,b r+b

r-2

ilL '1 -b--il I' ...[ ^A'Bl

.t

(a)

-A: B *"crD

B'[- (a) Dtl-c, +u, cr, . ,

tc

8l;[::];,.,{

'xl

br*b: bt*b2*I

I

'ur'

l),)

A.5.2-11

' 'uu;-.1

-1 Ir (c+u) f (c' -u) ]

-1 Ir(c+u) r(f-u) ] xY, (a; c+u, c t ; x, y)

L 1 .rc+f-2

F

f -2

x,y )

I

,c+ct -2

ffi,

14

4.5.2.15

(a,b,br;c+cr-I;

ffis(a,a' c+f>1

,b

'o

r,_.)!rJr,

Ir(cr+u)r(f-u) ]-1 ,,rft) {u,ur,.,b.i al* ,

'C2r,'Cri

ffi

-2

4.5.

2 . 16

-1

' ' 'xs)

xI, . ,xs)

2x'2Y)

]

"FSt) (b1, . ,br,

c,'''cr;d+u; x', , ' ,xr)

,c+f -2

ftatT-lT

Fo

(a,b;c+f-1,c' ; 2x,Y c+f>1

-2 -A:B;B,l- (a) rGif-DrC: D;D' lc, +f-I,c., ., z , (b) ; (b') ; '".. ..ll ' ^cr+f

cg: (d) ; (d') ; "^'")

cl*f ,

I

,c+ct -2

if .-. -=i; I l'I (' ; c*c ' ; 2x'2Y) c+cr >

1

[s) (bl, -.

.,b, ; c+f- I ; Zxr, . ,2x.)

rc;Trilo)-' c+f>1

,vjt) {.; .t*u,.2,: ,c x1

;

c+c, > , )c+t

4.5 .2.

c+f-I ;2x,Y) c+f>I

(a,b,b

ffiz

, )'_)

Ir(c+u; r(f-u) ]-i *Fa(i,,at,b,br;c+u;

(a,b,b' ;c+f-r ;2x,2y) , c+f>1

I(f-u)

,c+ct -2 b f (c+c t -I) Z' 1 \c+c' -t,,;.2x*2y1 c+cr > 1

atl-) 2''- '

-1

xl ' ' 'xs) Ir (cr+u)

-2

]

,ojt) tur, . ,br;c+u;

r(a;c+f-1,c';2x,Y) c+f>1

ffi

.rc+

xvr(a;c+u,ct-u;x,Y) II (c+u) I (f-u)

A.5.2.15

ffif "c+

A.5.2.12

221

,cr+f-Z

.

rc=If+ , ,a ^ rri'

(sr

)"'

(z

;cr'r- l' cr'' 2xl,x2,.,xs)

/r l 1

m

{"'o,'''b=i #.-Tlt'l" r ^cl*t-"c2' ' 'cs; 2*1,x2,.,x.)

cl*f r l

n.T-2 . . forrrtt;sr (br, .,b,,c,, .,c d+f

d+f>1

-l;2xr,. ,2xr)

S,

Integrals wifh resnect to Palameters

1r,

lA.s

))1

Single and Double Hypergeometric Functions

A.6.11

A.6 LAPLACE INTEGRATS

I

f (u) -1 Ir (cr+u) t (f-u) ]

i.5

.2 . 17

fi#tl"[') ^ .c+f-2

c+f-? 2- - |(-*J

,r[t) tr,rr,.,bu;c+u xr,.,xr) -l (c+u) r (f-u) ] .c+u,Ct;xr,.,xr) (c+c'|-r,br, br;c':u,cr-u; x1,. rxS)

'[l]u5"

-

',

1

]

(a'b,' ' 'b 5.; A.5.2.22 '[l]4" c+u,ct-u;xl,.,x s-) A.5 .2.23

Ir(c+u)r(f-u)]-1 . []].5" (a,a',Lrr,',b, c+uixl,.,xs)

Ir(cr*u)r(f-u) ]-1 A .'..r ,2 .24

.[].1"

(s)r- x It' ta'DI'''b,Ic+f-I; 2*,, . ,2xr)

rr

(a,a',bic1+u, c^...c :x,,.,x )

A.6.I.1

e-st

tP-

I r(t) dt;

r(c*t-D (t)'O Ld'-I'' '"s' c+f-l,c';2xr,.,2*k xk-]"'xs') c+f > I

iG.cJ)(1-2x,)'1" c+cr >

A.6.1.2

or

A.6.1.5

or,

2^c+ct -2

..(1-2x^)-bs s'

_lsl ('a'b .

p,

s>o

r'''

fotfit Iijtl" c+f>1 2'r'' '

Grr-D

c;xl s)

oFr(-;r;xt)

A.6.1.6

,Fr(a;c;rt)

A.6.1.7

,F, [a;p;xt)

A.6. 1 .8

oFl(-;c;xt+Y)

or(p;c;x/s,y)

A.6.1.9

lFI(a;c;xt+y)

o1(a,p;c;y,x/s)

AFB

(

[;] ;..*)

2

(r-+x1s2;P/2

A.6.1.5

A.6.

1.

1i

ci'Ir..

c+cl > I c,+f -2

(P+1) /

( c; xt) oF, -;

1 . 10

s2)

1F1

(p;

exp(x/s) zF

t(p,u;c;x/s)

(L-xlsJ *

. - D P+rn-l (al ';, ' '' m

. F^ ( A+m ts'

(b)

;mmx/sm) ;

b= i

c+ct-I;Zxr,. ,2xr)

c',

{-;$;xt2)

1

iGI-Il+;''

+f>l '

G-ax/ sz)-

rG;v/2;xtz)

A.6.1.4

A.6.

-

,rrtl,\;.;qx/

oFl(-;c;xt2)

I

?-,..-h

Ir (c+u) r (c' -u) ]

Ir (c+u) I (c'-u)

,1

c+f

.[]].5"(a,b,,,b,;

A.5,Z.Zl

_,. -b -2x,) -bt' ' (I-2xs)

c+f>1

Ir(c+u)r(f-u)]-1

L.5 .2.20

sPf(t) /r(p)

(1

ffi

*njs) {c*r-t,b1,. ,b,

Ir

A.6.1 L.aplace Integrals involving Single and Double Hypergeorhetric Functions

0

C+u;X1, . ,*a)

A.5.2.19

t',0;c,+r-I ,cz'' ' c=i2*l,xr,.,xr)

c1*f , I

x, ' ' 'xr) -1 Ir (c+u) r (f-u) ]

A.s.2.18

du

-F_')

^

,r[t) {",uic1+u,.2, .,.,

f(r.r)

(a'a';b,' ' 'b,

+f-I ,c2,' ,c.i2xf2,''x,

[f,]r

1/r (p)

n ,n+1,xlP,(a)' ^'r*r, q til (b) /

(- ; c; xt)

A.6.t.t2

*OF1(-;c';yt)

A.6.1.r3

oFt(-; c;xtz),

c+f-1;2xr,.,2xr)

(k)-(s), t;jua"'(a 'ar'b;

oF,

|

"oFt(-;c';yt-)

r

otl,f;.,c'

;4x/

s2

,qy/ 12)

tA.6

[.aplace Integrals

.t.14

!e -st aP-' a(.) dt;

sPf(t) /r (p)

Miltiple Hypergeometric Functions

A.6.21

sPf(t) /r(p)

p,s > o

,F,

(a;c;xt) F

,Fr(a';c';yt)

A.6.2.4

r(p,a,a' ;c,c' ;x/ s,Y / s) (fi,b,b'

F,

A,(r.I.15

;

c

;

x/s,r"/s)

(1-xls)-b(r-rl=)-o'

or(b,b';p;xt,yt)

(I

A.6.1.17

-xls)."

rF, Ia ;c;ys/ [s-x]

A.6. 2 .6 )

L 19

A.6.1.20

j') rrr, . ,br;p;

Y.{a; c, c' ; xt , Yt)

F

.---*.-=_-=-

. Y;[r)'(a;cl,.,cr; x- t...x tl l"T'

.*r'

bq':c xq t)

,F,

r 1]l

x FG*I "F.r2'

)'l 'xt) -F^ r L,'[gJ ; (

o(p,a;c,c' ;x/ s,t' / s)

4.6 .2 .7

- (r) ,, "= z lt1

d1' :f

(b-p,N)

(-N;b; st)

dt; p,s > o

,'P,l+9-b'(f)i*,=1 \t-p-b-N, is) ;"' "'

A.6 .2 .8

njt) tn,ur, 1,br;c;xr,/s, . ,xrls)

"1...(1-x./s).-b"r (1-x,/s)_h

xrtr. rx"t)

ojq) ru,

A.6.1.18 Ir..6.

oj') rrr,..br;c; *rt" 'xrt) . *

A.6.2 .5 A.6. I . i6

i" -st .P-1 f(t) 0

0

A.6.1.14

225

rrt

v'ia' tl

(c bt

.,oo

*1t'

xo t')

'

j') rt,o, YLt''

n[t) ru,pi.1,.,cri*l / s,.,xr/

[?]u50."

(p,br, .,bq,dr,

s)

.,d.i

c,f;xr/ s, . ,xO/s, Yr/s'''Yr/s)

f]r[t.'l

(p,c, r;b,, .,00,0,, .,0,

.,d

xrls,.,rO/r, Y'/s'''Y'/s)

r)

Y 'I

A.6.2 taplace Integrals involving Multiple Hypergeometric Functions A.7 HANKEL LOOP INTECRALS

sPf(t)/r(p) A.7.1 Hankel Loop Integrals involving Single and Double Hypergeometric Functiors

*jt)

oFl(-;c1;xlt)...

xOFr(-;cr;xrt)

2

,oFt(-,..,*ra') (rt ; ct; xlt)

,IF1(uricr;xrt)

.,cri

pjl.^ .(r).p ,rl,''! 'C t2'

oF1(-;cr;xrt2;

tFt

rn,c1,

..

.

xr/

s,.,xrls)

,1-Pr (p)

r(.)

A.7.1.1

oFr(-;c;xt)

A.7 .1.2

oFl(-;1-p;xt)

t/2nil(o+)"st t-P r(t) dt;

^

A.7

.1.3

or,(-;c;xt2)

A.7

.t.4

or,

t-;f;*t2)

exp(-xls) z,

{*,tl:c;-qx/s2)

(t*4x1 t2yP/2-t

s >o

l

Hankel Loop lntegrals

226 'I

s^-nrr(p) oF,

A.7.1.6

,Fr(a;b;xt)

4.7

.t

.7

A,7.I.B A.7.1 .9

,F,

(a,1-p;b; -xls)

4.7 .L.22

F, (

,r, r-i,fi

,arbix/t)

i sx)

a;-;x/t) 2Fo(-n,

(-n,a;p; sx) ,F,

4.7.I.11

2Fo(-n,p; -;x/t)

A.7.r.13 A.7.1.14

A.7.1.Is

A.7.1.16

,ro(-",f

2F0(-n ,ai-;x/t

oF1(-;c;xt2)

4.7 .t.17

*oFt(-;c';yt-)

A.7. i.18

,Fr(a;c;.rt) "1F1(rt;ct;_v-.t)

A.7"1.19

h ,7 .1

!t

.20

-;x/t) ,b;-;y/t)

,Fo(-m,a;

xrFo(-n

or(a,b;c;xt,y)

o, (b;

A.7 .L.24

02(b,br;c;xt,yt)

F, (1 -p,b,b I ;c;

4.7 .1.25

0.(b,bi;1-p;xt,yt)

r(-n;p/2;s?

A**FB

;

;

-x/s,y)

y, xs)

-xls,

-y / s)

_h _hr (1+x/s) " (l+y/ s) "

(

1-n m-D - -n t" 'E' ';t;LgJ--il

F

4(+,t-!:c,c'

F

,{l

ir**, #J

;4x/ sz ,qy/ t2)

t)

A.7 . t .27

Y. (p ; c ,c'

4.7.1.28

Fl(a,p,b;c;x/t,y)

o,

L71?.

F, (n,

or(b,bt;c;sx,ys)

L

b

;x/t

F

4.7

,

y/t)

.t.34

-n,b,b';1-p;

,F, (a;b;xs) rF, (a';b';ys)

yt)

,Fo(-m,b;-;-xls) I, ^F^f-n.b':-:-yls)

Fo(p,b;c,.tti,p

V.(b;c,cr;xs,ys) Z

xt,

I

2/Dorr{-

oFr(-

A.7.1.35

-p ,a,a t ; c, c t ;x/ s ,y / s)

A.7.1.36

v*(a,b;1-p,c;y,xt)

A.?.I.3?

F4(p,p;p,p;x/t,y/t)

F. ( -rn, -n, a, b; p; xs , ys )

(a,1-p,b;c; -xls,y)

(a,b;c;xs;y)

Yi(a,b;c,c';y,xsJ

r(a,p,b; c, c' ;i,y)

F3 ( -m,

i

-y/s)

oF1(-;c;xs)oFr(-

Fr(p,a,a';b,b'; x/t,y/t)

.t.32

4.7.I.33 4.7

,y /

,b' ;c;x/.t

x/ 4)

AFB*r( Df,u (b),#,.,L#a; _. - otu'o**_r n

F1

c

I

Fo(a,1-p;c,cr;

A.7.1.51

oru(if;]i*2.*)

dt;s

F, (a; c; xs+y)

o, (p, b; c;x/ t ,y)

)

orur[f;]:*t't

f(t)

Fr(a, t-p,b;c,c

4.7 .1;26

P+1 2 ,.thl(-ni , is xl+) ,F

,

r,c;Y,xt)

-n t1-sx.)

;-;*it2)

,2.

01(a,p; c;y ,x/t)

.-P

(-n;b;sx)

4.7.1.10

,p/2;-;*/t2)

fi1to.l.st

227

-t -23

A.7

1+x,/s) -

,F,

,Fo(-n

r1-Pr6p1 r1.1 4,7 .1.21

,Fr(-n,p;b;x/t)

4.7 .1.t2

Sin$e and Double Hypergeometric Functions

il;7l* flr-

(a;1-p;xt) lF1 (-n

A.7.lI

/tzni1(or).st , -P r(t) dt; s > o

f(t)

.) (-;l-p/2;xt-)

A.7.1.5

lA.7

(I

rx/ s) -'rr, {r, U ;. ;}-*)

xs+/s OF,

(_

2-

;p;xys

.)

o

Hankel Loop Integrals

2;lt

I^.7

A.8.1l

Single and Double Hypergeometric Functions

229

A.8 MELLIN INTEGRATS

A.2,7 Hankel l.oop Integrals involving Multiple Hypergeometric Functions A.8.1 Mellin Integrals involving single and Double Hypergeometric Functions

L1)1

4.7 .2.2

L.7.2.4

,1-Pr1p1 11.;

I ;(O+) c-st t-p f(t) n-il

ojt) rrr, . ,br; r-p; xrtr. rxrt)

_h 1t+xrls) "1. . .(l

o)'J tu,,.,br;c;

xrt, . ,x t)

vj') rn;c1,.,cr; xr/t,.,xr/t) vj')tu;c1,.,crr x1t,.,xrt)

SrO r jx

f (x)

*xrls)._b"r A.B.t.1

AFB(

x

rj') {1-n,rr,.,u

-*1 s

[;] : -k") (

t_

o

Il

o

nl'lt-"r,.,-nr, 4.7 .2.6

bl' ' 'br;1-P; xrt, ',xrt)

';'; r* j [fi]r

arg , | .[tzn*2n-r-l) A.8.1.2 x

_*l

-(r) r;'(P,b;c1,.,cr; xr/t,. ,xr/t)

-

-xr /s')

,FO(-n.;br;

4.7 .2.8

4.7 .2.9

4.7 .2.tO

rf'l tr,ur,.

,br;c;

xr/t, . ,xr/t) nFr(-;cr;xrt) "oFt(-;cr;xrt)

iTf (l -b. -s)'i=n*1J If (a.*s) i=m.l J

Y

A.8.1.4

tF1(ul;cr;xrt)...

,1F, (ar.;cr;x.t)

x

r(a;c,c';-hx,k) Co

-sr, . ,a.-s, > o

r(br+ur)r(fr-ur).. xl(b +u )I(f -u ) 'rr--rr:

>o

r(ar+fr)..r(a.+fr) lI

I

r

r

,ol") (.r*rr, . ,..*f xr/2, . ,xr/2)

r(b1+u1)r(fr-ur).. r(br+fr)..r(br+f") xf(b +u ll(f -u ) .b, +f,L+. +br +f r 'rr"rr' tI

o

r(r),- Ldlr',d,rLr o2 -P1*1

*f(uI,. ,u.)dur. .du,

t(ar+ur)r(fr-ur).. xl(a +u lfff -u ) 'rr,.rr' ,ojr) [ar*u1, . ,ar+uri b;x,,. ,xr)

al*fl,.,ar+f.

P1*I,';-Prxr) P1't1 ,. ,Pr, ta ' 0

A.9.5.4

4.9 .6.2

> -1/2

-ptxI

*5",i ;C1'. A.9.5.3

fF{:-rrf-

Pr*r) P1,s1,.,Pr,sr > O ,F,

'dx1 " 'dxr

r(c)f(s1)..r1sr)

oF1(-;c;-plx1-. A.9.5.1

f(ur, . ,ur)

oi,. rroo.oi,i. 1,; .ir1*,, . ,*") -

rr,t,-"/l'_,].,., /;,_r:

-\

r(br+ur)r(fr-ur).. xf(b +u lr(f -u ) -rr"rt' ,Ff') {.,br*rr,., br+ur,;br+f1

''' b.*fri*1,.,xr)

l'rl ,Fi'' (a1,.,ar,b1*f,,.,br*f, xr/2,.,xr/2) b1*f1,.,br*f., o, lx. I..1

I(b1+f1)..r(b.+fr) zbrlft *.*br*f,

'rf')

{",br*f1, .,b.*rr;

cl,.,cr;xl/2,.,xr/2) b1*f,,.,b.*fr, O,xlxrl.r

r(br+fr)..r(trr+fr) lLE_;iltE=trI r r x (l-xr/2- . -xr/2)-a bl*f,,. ,br+f. > O

Multiple Integrals

.) {H

lA.e

- ],+i-

{zrD-r lrtr_i_.(r)

f(ur, . ,ur)

-ul , . ,br-ur; c; xI, . ,xr) u1*b1,. ,ar+b, > O

b1

1

r[') t.,u;cr-u1,.

,

'''*.)

a1*r1, . , cr*r, )

vj') {.;"r*rr,

lntroductionThe list of fifty computer programs which follows consists of representative examples appropriate to the evaluation of the hypergeonetric integrals tabulated above. It is taken that all quantities are real unless otherwise indicated. For obvious reasons, the number of parameters and summations has been limited, but the form of the programs is such that they may easily be extended to cover even the most complicated cases cf the integrals under consideration. The internationar computer language F0R.TRAN lV is employed.

a+b

rI *Fi a-+b-+1 arr'+b +l -t1 2 "', 2

x1,.,x.) l*al.r

2.1*f1* .*cr*fr-2r

" /.(r) ./f("r,. ,r.)dur. Fi

(a,b; cr+f,

.du,

rr+f

.

.

It is suggested that these programs are run for a 1ow value of lrl, say ltl=S, an.C then re-run for lrl=6. By this means, a practical indic_ ation of the speed of convergence of the summations will be obtained and ]tl rnay be i-ncreased untiI. the desired degree of accuracy is achej,ved. It must always be ascertained that the series, single or multiple, being investigateC is either convergent o.- a suit_ able asymptotic series. Note. The symbols rm'and'n'in the READ and WRITE orders should be replaced by the appropriate numbers for the input and output channels on the equipment being used.

,2xr) . ,u

B.

cr+fr-1,.,cr+f.-1; 2xr,.,2x,) rlxrl.l/2

c.+fr- 1 ; cl+ul, tr**ri*l ' ' '*r)

".*t.i*1,.

t1 *b1 "1 (r),*1

rf') tu,rr,

r[r) {",cr*rr-r,

A.9.6.10

Computer Programs

r(a,+b,)..I(a r- I r +br') 2uI*bl*.*".*b.

2x1 , ,

rf') t.,u,, . ,u x,,-,x 1T'

.

.

>

al*f',.,c.+f.

tt*'ri*I

B

f (c, +f, -1) . . f (c_+f_-1) llrr

xl(cr+ur) f(f1-u1) xf(c +u )f ff -u I 'rr"rr'

A.9.6.8

.

xlfa +u lf(b -u ) 'rr"rr' *n[') {rr*,.,1 , .rar+ur,

f (ur, . ,ur)

A.9.6.7

,_

xf(ur,.,ur)dur..du.

r(ar+ur) r(b1-ul)

A.9.6.6

Y +i-

./r,

I

Programs for the Evaluation

n.r.r .

,

,

,*r)

(1-2xr- ..-2xr)-a

vjt) tr;cI*fl-1,. ,c.+fr.-l 2xr",2xr)

ffiS

of Euler Integrals

j, "" ' (t-u)b-l zFzGt,.2)dt,dr;ux) du, a,b > o.

(m,6) A ,B ,CI ,C2, Dl , D2 , X I F (A) 8,8,9 6 FoRi.rAT(7F 1.2) I READ(rn,7)1.{ READ

;

7

FoRlrAT(r2) CALL F(A, B,C1,C2, D1,D2, X,l,1,S) WRITE

(n, 5)A,B,CL,C2,DL,D2,

FpRI'1AT(13H PARMETERS GOTO

I

=,

X,t\1,S

7F5,2/3H I,l=,I2,SH F=,1PEt4-6J

STOP END

SUBRoUTINE F(A,B,C1,C2,Dt,D2,Xit,t,S) S=O.

O

T=1.O [email protected]

1

N=1,1.1

AN"FLOAT(N)

B.lI

Programs

,I

n.r.sfrffiJo,"-'

-I.o

S=S+T

T=T* ( (A+AN)/ (A+B+AN) )

I

lB.l

Programs for Euler Integrals

240

*( (Cr+AN)/(D1+AN)

)

*( (C2+AN)/(D2+AN)

J

CONTINUE RETURN END

READ(m, 6) A,

rF(A)8,8,9 (r-...,)b-1 ,1 t u2

t

i)l',_,:,, ig2,h2; ux, (1-u)y

du,

B,C, D1 , D2

, G1

,G2,Hl ,H2 ,X,Y

(m,7)

Irl

I2)

c0'i'0 |

Z

=,

13F5

.2/3H M=,12,ZH F=, 1pEt4.6)

1

STOP END

F(A,B,C,DI,D2, D3,G,H1

D0 1 N1=1,M a,^i 1= FLOAT (N

I)

A1=A+AN1-I.O

BI=A+B+ANI-I.O 5,! =Q+ANl

STOP

-1.0

Gl=G+ANI-LO

ENb

T2=Tl

SUBROUTINE F (A, B,C, DI, D2, G1, G2,H1,H2, X,Y,II, S) O

G2=G1+AN2-l.O T3=T2 D0 3 N3=1,M

-t.O

T2=TI

ANS=FLOAT(N3)

D0 2 N2=l,ltl

A3=A2+AN5-l.O

AN2=FL0AT (N2 )

B3=B2+AN3-

C2=Ct+AN2-1.0

G5=G2+AN3-1

S=S+T2 T.2=T2* (C2/ A2) * ( (D2+AN2 - 1 .0)

T2=T2*( (B+AN2.1.o)

/ (G2+AN2- 1 .o) ) / (H2IAN2-1.o) ) * (Y/AN2)

CONTINUE

Tl =T1* (C1lAi j * ( (Dl+ANl T1=Tl * ( (A+ANI-

l.

O

C3=C2+AN3-L.O

A2=Al+AN2-l.O

RTJTIJRN

FL0AT (N2 )

C2=Cl +AN2-1.O

Cl=C+AN1-1.0

CONTINUE

=

B2=Bl +AN2-1.0

1 NI=],M [email protected] (Nl) [email protected]

41=4+B+ANt

D0 2 N2=I,I{ /tl{2

A2=AI+AN2-l;O

T1=1.0

I]ND

J

S=0.0 T1=1.0

CALL F (A, B, C, D1, D2,Gl,G2,HI,H2, X,Y,IvI,S) I{RITE(n, 5)A, B, C, Dtr,D2,Gl, G2,H1,H2, X,Y,S [email protected],{AT ( 131'{ PARAI\IETERS =, 1 1f5 . 2 / 3H t'l=, T2, 3H F=, I PE I 4 . 6)

S=0.

FoRI'{AT(13H PARA}{ETERS

SUBROUTINE

6 [email protected]\I,{T(11F4.2) 7 [email protected]'lAT (

du, a,b > O.

I

7 [email protected]\AT ( I2) CALL F(A, B, C, Dl, D2, D3, G,HI ;H2,H3,X, Y, Z, [{, S) WRITE (n, 5 ) A, B, C, D1, D2, D3, G, Hl,H2,H3,X, Y, Z, I',I, S

c0r0

a;b > O. READ (m,6) A, r F (A) 8,8,9

B,C, DI,D2,D3,G,HL,H2,H3, X,Y,

i

6 [email protected]\,AT(13F4.2) 9 READ (m, 7) 11

I

8.1.2 ,tfffir [ ,"-1 ro " ,l;

241

(l -r) b- I

idji ,r "Fl :1 [''dr;dz lg:h-.;h, ;h.; ,*,,,r,,,

)

T=T*(X/ (l.O+AN))

9 READ

for Euler Integrals

1

.O)

/

/

(Gl+AN1 - 1 . 0) ) (Hl+AN1- I .O) ) * (x/ANl) - 1 . o)

.O

5=5+T3

T3=T3* (A3l83) * (C3lG3) * ( (D3+AN3-

1 ;

0) / (H3+AN3-1 . O) ) * (z/AN3)

[email protected]

T2=T2*'(A2/ Bz)* (C2/ c2) * ( (D2

+AN2 -

I . o) /

(H2 +AN2 -

I . o) ) * (Y/AN2 )

CONTINUE

TI=Tl * (A1lB1) * (Cllcl ) * ( (D1+ANl - I .0) / CONTINUE RETURN END

(H1+AN1 - 1

-

o) ) * (x/ANl)

hojnrnr

242

r].r.4 I

for Euler Intcgralr

lB.l

READ(m,6)A,B,C, D,GI,G2,Hl,H2,X,Y F

(A)

5

8,8,9

T1=1

I

=,

1OF5

.2/sH

M=

[email protected]

,I2

,3H F=,1PE14.6)

I,l,

s)

F

(A, B, C, D,G:,

G2, H1,

I

H2, p, QI, Q2, x, y, z, u, s)

.0 Nl=1,1,,1

P1=P+ANI-1.O

F(A,B,C,

D, G1,

T2=Tl

G2,H1,H2, X,Y,M,S)

[email protected]

S=0.0

N2=l ,M

A2=A1+AN2-i.O B2=B1+AtrI2-I.O

(Nl )

L

2

AN2=FLOAT (N2)

T1=1.0 D0 I N1=1,1'l

P2=PI+AN2-1.0

O

T3=T2

81=A'B+ANl-1.0

D0

T2=T1

D0 2 N2=1,lvl

A3=A2+AN5-1.O

83=82+AN3-1

A2=At+AN2-I.O

.O

S=S+T3

B2=B1+AN2-I.O

T3=T3* (A3/83) *( (C+AN3-1

S=S+T2

T2=T2* A2* (A2/B?)*( (C+AN2-1.

o)/

CONTINUE

T1=T1*Al* (A1/B1)

T1=T1 * ( (Gz+aNr

3 N3=l,I,l

AN3=FLOAT (N3)

AN2=FLOAT (N2)

-t

(D+AN2-1. o) ) * (y/AN2)

(Gl *6q1 -, . O) / (rII+ANt - I .O) .0) / (H2+ANr -1 .0) ) * (x/AN1)

n(

l

.0)/ (D+AN3-t.o)

CONTINUE

T2=T2* (A2/BZ) *( (G2+Rtqz-r

T2=T2* (Y/A.\2)

)

* (z/ANs)

.q /p2)* ( (H2+AN2-1.0) / (Q2+AN2-1.o)

)

CONTINUE

TI =Tl * (AI /B1 ) * ( (CI

CONTINUE

T1=TI* (x/ANI)

RETURN END

- 1 . o)

/pt)

* ( (H1 +AN I -

l

/

(Ql +ANt _ r . o) I

. .l;i [*'n;'i"nt',*.,,y,,,-l

du, a,b > O.

+tr1r,11

.

o)

CONTINUE RETURN END r7

s r[i+h ,ol" a-l x

(r-r)b-1

,F,

(c;d;uz)

-o:2;21--t*r'hrlEr;hr; '1'l;I

I

lo, o, i 9z ,'*'"_l

RIIAD(m,6)A, B,C,D,Gl ,c2,HL,H2,p,Q1 I p (A) 8,8, e FoRMAT(14F4.2)

6 9 RBAD(m,7)M ? NORMAT(I2)

y, z,

A1=A+ANI- 1 . O B1=A+B+AN1-1.0

[email protected]

I

X,

nNI=FL0AT(N1)

END

B.,

Ql, Q2,

1

suBRoU"riNE

STOP

ANI =FLOAT

p,

S=0. o

(n, 5)A, B,C,D,G1,G2,HI,H2, X,y,M,S

AI=A+ANI -

Gl,G2,Hl,H2,

END

FORMAT(I3H PARAI,{ETERS GOTO

8

B, C, D,

243

(n , 5 ) A , B , C , D , G I , G2 ,l{ I , H 2 , p , I , y Q Q2 , X , , Z , l"t, S FORMAT(f3H PARAMETERS -, t4F S.Z/ia'tt=,Iz,3H-p=,tprr+.o) GOTA

CALL F(A, B,C,D,G1,G2,HI,H2, X,Y,T,I,S)

5

(A,

Intcgrals

8 STOP

6 I.:oRMAT(10F4.2) 9 READ (m,7) lit 7 FoRMAT(r2) WRITE

F

WRI TE

* zFz(8i,8r;h'hr;ux) du, a,b > o,

I

Progrtmr for Eulcr

CALL

(r-u)o-' rrr(c;d;uy)

u,r##1,, "-'

B.1l

8.1.6 du, arb >o.

','-' (l-,,):-1

r!99: 'o f

,F, (c;d;uw)

Lprqr;qz;qs;

,e2,X,\ ,Z READ (m,

I

F

6) A,

(A) 8, 8,9

B;

6 [email protected]'{AT ( 16F4 .

9 READ (m, 7J

M

C, D, G,Hl, H2,HS, p, Ql, Q2, QS,

2)

J

X,y,Z,W

244

Programs for Euler Integrals

B rl

lB.1

Programs for Euler Integrals

FoRMAT(r2) B, C, D, G, Hl,H2,H3,P, QI, Q2, Q3, X, Y, Z, W, l.{, S) . 5 ) A, E, C, D, G,Hl,H2,H3, P, Q1, Q2, Q3, X, Y, Z, W,!{,S [email protected]( I 3ll PARAT''IETEp5 BF 5. 2 / 2X, 8F 5 - 2 /3H y=,I2,3H F=,

CALL F (A, WRITE (n

8

=,

I

GoTO

uu-l (r-r)b-l ,r, i.r,cr;d;ul,) * :F2(B' Zz,Es;h.hz; It-u]x) du, a,b i READ(m,6)A, B,U.,C2,D,Gl,G2,G3,Hl,H2,X,y -1 < x'y <

IPEI4 . 6)

STOP END

STTBROUTINE

F

S=0. o

.Q

B1=A+B+AN1

-1.0

7

8

N2=1 ,M

D0 1 N]=1,I{ .r'\i=FLoAT(Nl) Bi-A+B+ANI-l.O

si:Filtil,-i:s T3=T2

T2=T1

3

N3=1.r1 AN3= F LOAT (N3)

[email protected]

82=Bl+AN2-l.O S=

T4=T3

2

N4=1,1,{

A4=A3+AN4-1.0 B4=B3+AN4-i.O

1

T4=T4* (A4 /84)* ( (C+AN4-1.0)/ (D+AN4-1 .o) ) * (W/AN4) CONTINUE

/

CONTINUE

T2=T2* (A2/82)* (G2/p2)* ( (H2*41s2-1.o)

(Q3+ANS- 1 . o) ) *

(z/ANj)

/

(Q2+AN2-1. O) ) * (y/AN2)

-'l'l * (A1 /B I ) * (G1lPl) * ( (u1+61s1 - 1 . o) / l(:sNI'rNrri

(Ql+AN1 - 1 . o) ) * (x/ANr )

1'l

\2=T2*((C1+[II2-1 -o) / B2)* ( (C2+,{\'2-1. o) / [D+AN2-1.9; I T2=T2* ( (A*RwZ-t - O) IANZ) *Y CONTINUE

((Gr*451-, . o) /Bl)

=T1 * ( (G3 +ANI CONTINUE

Tl

1

6 9

*- -"*-**- \r

/ (H}+ANl_i . o) )

nll*/r.erANrr , \\u,

I vJlru\t]"i /^rr1\&\, ^\

,I

^ r [a+b) D'r'o il;jl{itr

Hli'rrI{N llNl)

* ( (G2+AN1 -1 . o)

- I nl /fH"+AN'l-l

RETURN END

S=S+T4

CONTINUE

S+T2

T1=T1*

ANa=FL0AT (Na )

T3=T3* (A3lB3) * (G3lP3) * ( (H3+AN3-1 . o)

2 N2=I,Il

A\2=FL0AT (N2J

A3=A2+AN3-I .0 83=B2+AN3-1.0 G3=G2+AN3-i.O P3=P2+AN3-I .0

4

STOP

Tl-l.O

B2=B1+AN2-L .0

4

F=,1pEi4.6)

1

S=O.0

A2=Al+AN2-I.0

D0

cora

suBRoUTINE F(A,B,C1,C2,D,G1,G2,G3,Ht,t12,x,y,Nf,s)

[email protected](N2)

D0

1.

E.\D

T2=Tl

2

>o

FoRrlAT( r2)

e r,q, B, cr, c2,D, GL,C2, G3, H1, H2, X, y, [r, s) 9t!l IITRITE(n,5)A, B,C.1,C2,D,G1,G2,G3,Hl,H2,X,y,l,,l,S s F0RMAT(1sH PAR-{TTETERS =, 12F5.2/34 M=, 12,sH

GI=G+A\rI-I.0 Pt=P+AN1-i.0 [email protected]

Jo

rF(AJ8,8,9

TI=1.O tr]=tr+{tr}l=l

I

6 FaRMAT(12F4.2) 9 READ(m,7)1",t

(A, B,C, D, C,H],H2,H3, P,QI,Q2,Q3, X, Y, Z,W,M,S)

[email protected] 1 N1=t,lr{ ANI - FL0AT (N1 )

.I

r (a+f) I (a) r (b)

8.1.7

245

READ (m, 6)

{

'0 I

trf "'8r'ht; EThr; I du, a,b vO, ^. .1:2 ux,uy,l ''0:2;21 qI:Pz,q2i J., . L-'Pr **, ,,

CONTINUE

;h

,-"I

Prograrns

fDa TD.L

for Definite lntegrals

B.2I

T3=T3*(A3l83) * (E3/ {P2+AN3-1.o) ) * ((H+AN3-1.o) /(Q2+AN5-1.0) ) T3=T3*Z* (W/AN3)

3 CONTINUE

T2=T2* (AZ / B2) * (E2 T2=T2*Z* (V/AN2)

/

(P | + AN2 -

I . o) ) * ( (G+AN2- 1 o) / fQl

for Definite Integrals

253

P3=P2+AN3-1.0 T4=T3

4

D0

I . o) )

N4=1,M

AN4 =FLOAT (Na

)

P4=P3+AN4-l.O

CONTINUE

I

+AN2-

Programs

TI =Tr * (Ar/Bt ) * ( (C+ANr

-r

/

.O)

(D+Arrir -

S=S+T4

r. o) ) * (u,/ANI ) *Z

CONTINUE

4

RETURN END

T4=T4 * ( (E3+AN4 - r . o) / P 4) * ( (c3+AN4 - I . o)

/

(Q3+ANa - 1 . o) ) * ( x/Ar,i4)

CONTINUE

T3=T3* (As / B3)* ( (Ez+ANs- I . o) / P 3) * ( (G2+AN3T3=T3* (W/AN3) *Z

i . o) )

1 . o)

/

(Q2+AN3-

1 . o)

/

(Q1 +AN2 - 1 .

3 CONTINUE

az -^ [I ''0

8.2.6

z

2* (A2 / 82) * ( (E 1 +AN2 - 1 . o) / P2) * ( (G1 +AN2 T2=72* (V/AN2) *Z T

a-l z

I

x

,Fr(c;d;uz)

gl;e2,gztes,g3; I ^o:2 ttrl l-'"r, v:'wz'xl I

L!'0, ;ez;e3r a >

I 6 9 7 5

READ(m,6)A,C,D,EI,E2 , E3, Cl I F (A) 8,

8,9

2--T

2 CONTINUE

T1=T1 *

dz,

1

(A1lBl) * ( (C+AI.I1 -1 . o) /

(D+ANI - I . O)

)'t (U/ANi ) *Z

CONTINUE RETURN

J

END

O.

, G2 , G5 , P , QI , Q2 , Q3, U,

\',1{, X, Z

az

8.2 .7

-al

FoRII/\T(r2) CALL F (A, C,, D, E1,E2,,E3,G1, G2, G3, P,Ql, Q2, Q3,U,V, W, X, Z,M,S) l\'trITE (n, 5)A, C, D, E1, E2, E3 ;G1,G2,G3,P, Q], Q2,Q3, U, \" tt, X, Z,M,S FORIvIAT ( 1 5H PAMMETERS =, 9F g : 2 / 2X,9F 5 . 2 / 3H l'l=, I2, 3H F=, 1PE14 . 6)

I

8 STOP END

z

a-I

I

)

FoRMAT(18F4.2) READ(m,7)lt{

GOTO

U

"

2F

r(.1,.2

;d;uz) sFz (.r

a > O, -l < uz,vz <

,e2,e3;g,g2ivz) dz,

L

1 READ(m,6)A,C1,C2 ,D,El ,E2,ES,G1 ,G2,tJ,V ,Z IF(A)8,8,9

6 9 7

FoRMAT(i2F4.2) READ(m;7)M [email protected](I2)

CALL F(A,C1,C2,D,E1,E2,E3,G1,G2,lJ,V,Z,l{,S) SUBROUTINE F (A, C, D,E7,82, E3, G1,G2,G3, P, Q1, Q2, Q3, U, V, W, X, Z,I{, S) S=O.

O

D0

I

Nl=1,M

(n,5)A ,CL ,C2 ,D ,EL ,EZ ,E3,Gl ,G2,lJ ,V ,Z ,ll,S [email protected](13H PAMMETEpS =,12F 5.2RH M= ,I2,3H F=, COTO I WRITE

5

T1= 1 .0

AN1=FLOAr(N1)

B

1pEi4.6)

STOP END

A1-A+ANI-1.0 B

SUBROUTINE F (A, C1, C2, D, E1,E2,83, G1, C2, U, V, Z,IU, S) S=0. o

I =A+ANI

t'2=TI

D0 2 N2=I

,M

T1=I

I

.O

Ni=l,M

AN2=FL0AT (N2)

D0

A2=A1+AN2-1.O

ANI =FLOAT (N I )

82=Bl rAN2-1.0 P2=P+AN2-1 T3=T2

D0

.O

3 N3=l,M

A1=A+AN1=1.O B1

=A+ANI

'12=Tt DO 2 N2=T,M

(N3)

AN2=FL0AT (N2)

A3*A2+AN3-1.0 B3=82+AN3-1.0

A2=A1+AN2-1.O

AN3= rLOAT

o) )

82=BI+AN2-1.O S=S+T2

--4

254

Programs

T2.T2* (A2

2

I

/

for Delinite

[B'2

B.2l

/

(G1

CONTINUE

for Definite Integrals

Programs

255

S=S+T3

+AN2: I' O) ) *V*Z (G2+AN2 - 1' O) ) * ( (E3+AN2 - 1 . O) /AN2)

B2)* r 1[ | +AN2 - I' O)

T2.T2* i (rZ*nirz- r . 0) /

Integrals

T3=T3*(A3lB3) * (E3l (P2+AN3'1.o) ) *( (G2+ANS-I.o) / (Q2+AN3-1.o) T3=T3* ( (HZ*4113- 1 . O) /AN5) *lr'tZ

)

3 CONTINUE

. 0) / (D+ANI-1.o) ) TI =T1 * ( (C2+ANI - I . O)7AN I ) *U*Z

Tl=Tl* (Al/Bl) * ((C1+AN1-1

(AZ / 82) * (82l (P2+AN2 - i . o) ) * ( (G1 +AN2 - r o) " T2=T2* ( (Hl+612-1 .O) /AN2) *V*Z

T2=it2*

CONTINUE

2

RETURN END

. I

/

(Qi +AN2 - 1 . O) )

CONTINUE

T1=T1* (A1/Bl) " ( (Ct+41t11-1 . 0) T1 =Ti *U*Z

/

(D+AN1 -

I

.

O) ) * ( (C2+ANr

:r

.

O)

/AN1l

[email protected]

RETURN END

8.2.8 dz,

u ,-^

8.2.9

[ ,^-1 ,Fr{r, ,cr:'d;uz) r ,q,r: I ES:O lP vz,wz ,xzl dz , ''o:2 'gt,hr;E,hr;E=,hr; L J

a > O, -I < uz, (v+w)z < l'

;

I

I READ(m,6)A,C1,C2,D,E,GI,G2,Hl,H2,Pl,P2,Ql,Q2,tJ,V,W,Z

I

IF(A)8,8,9

a > 0,

6 FoRr.rAT(17F4.2) 9 READ (rn, 7) IrI 7

5

(A,C1, C2, D, E, G1, G2,H1,H2,P\, P2,Q1,Q2,U, V, l',l, Z,l"l,s) WRITE (TT, 5) A, Ci, C2, D, E, GI,G2,HI,H2,P 7, P2, Q], Q2, U, V ; W, Z, M,S FpRMAT(i3H PARAMETERS =, 9F5 .2/2X,9Fs.2l3H M=,I2'3H F=,1PE14'6) 7

8 STOP END

SUBROUTINE F (A,C1,C2,D,E,GL,G2,HI,H2,PI,P2,QI,Q2, S=O. O

T1=1.0

D0

I

Igtii;:)i;'

"

7 nOnu,nr i r 2l

(A, C1, C2, D, G1,G2, G3,Hl,H2,H3,p, Q, R, U,\r, W, X, Z,I.{,S) D, G1, G2, G5, H1, H2, H3, p, Q. R, tJ, V, W, X, Z, l,{, S 5 FpRMAT(I3H PAMMETERS =, 9F 5. 212X,9F5.2/3H l,l=,L2,3H F=,lpEt4.6)

CALL

F

WRITE

(n, 5) A,Cl,C2,

G0r0 |

5 STOP END

Al=A+ANI-1.O B

U, V,W, Z, M, S)

6 9

I =A+AN

I

T2=T\

D0 2 N2=1,M

At{2=FLoAT (N2)

SUBROUTINE S=0. 0

F

(A,Cl,C2, D,Gl,G2,G3,Hl,H2,H5, P,Q,

T1=I.O

D0

I

N1=1,M

AN1=FLOAT (N1 )

4!=[]+ANi-1.O

ti=ttffil '

B2=BI+All2-1.C

T2=Tl

E2=E+AN2-I.O

D0 2 N2=1,M AN2=FL0AT(N2)

T3=T2

D0 3

N3=1,NI

ANS=FLOAT (N3)

A5=A2+AN3-1.0 B3=82+AN3-1.0 E3.ll2+ANS- I .0

"_***-!t*

Z

Nl=I,M

Alr r =FLoAT (Nt )

h*

6) A, CL,C2,D, G1, G2, G3,H1,H2,H3, P, Q, R, U, \:, l.{, X, rF (A) 8,8, e

F

GOTO

-1,<

READ (m,

FORMAT(I2)

CALL

I

A2=A1+AN2-1.0 B2=B1+AN2-I.O P2=P+AN2- I . O Q2=Q+Nrtl- 1 . g

R2=R+AN2-1.O

R, U,V.,W, X,

Z,M,S)

'13

u1.3

Progrmr for Repeated Intcgrdr

256

B.3l

"'l'2

[email protected]

t)/ 3 N3=1,M

L NI=l,lr.l

BN=FL0AT(N)

A3-A2+ANS-l.0

S=S+T

113=82+AN3-1.0

T=T* ( (A*AN)

I)3=P2+AIri3-1.0

7='1* (U/ (AN+

.0

1

R3=R2+AN3-l .0

(A+BN+AN) ) * ( (Cr *All)

/ 1

.0) ) *Z

I (or *AN) ) * ( (CZ*41.11 7 (D2*aN; I

CONTINUE RETURN END

T4=T3

D0

a o.

1 READ(m,6)C1,C2,Dl,D2,GL,C2,HI,H2,x,Y IF(Cl+C2)$,3,9

'h-

Programs

260

for Barnes and Related Integrals

lB.4

B.4I

I

[email protected]

1 N1=l ,lr1

ANl=FIOAT (NI )

WRITE (n,5) CL ,C2,DI ,D2 ,GL ,G2 ,HL,H2, X, Y,M, S [email protected](f 3H PARAI{ETERS =, 10F5. 2/ 3H U=,I2,3$

rt=

(CL+C2) /

2.o

E2=E1+.5

F=, 1PE14.6)

A1=E1+ANl-1.0

1

B1=E2+AN1=l.O

STOP END

T2 =T1

Dfi 2 N2=l

S=O.

O

TI=1

.O

,M

(N2) A2=A1+AN2-1.0 82=81+AN2-1.O

F(C1,C2,Dl,D2,G1,G2,HI,H2;X,Y,M,S)

SUBROUTINE

[email protected]

D0 1 N1=I,M

T3=T2

AN1=FLOAT(N1)

D0 3 N3=1.M

A1=CI+Gl+ANl-I.O

AN3=FL0AT (N3)

T2=TI

A3=A2+AN3-1

.O

D0 2 N2=t,lt

B3=B2+AN3-1

.

(N2) A2=AI+AN2-1 . O

S=S+T3

AN2=FLOAT

S=S+T2

T2=T2* (A2 CONTINUE

/2

.

O) * ( (G2+AN2 - 1 . O),r (H1 +AN2- 1

=T1* (A1/2 . O) * ( (C2+ANI

-r

O)

.

/

.

0) ) * (Y/ (H2+AN2 - I .o) )/Purz

(D1 +ANt - 1 . o) ) *

(x/

(P2+!N2-l .o))* (82/

CO;'ITINUE

Ti =T1* (A1/ ( P1+AN1 - i. . o) ) * (Bl

(D2+AN1 - I . o) )/AN1

(Q1+ANI -

1.

o) ) * (x/AN1)

RETIJRN

,c, +cr-1

(

?ll i'(cr +c.t)

t

,"

i"q

", , I r(c.,+u) f (cr-u)

I

-'*

*orro 'O:2

1 READ(m,6)C1 ,C2,P1,P2,P3,QI

8.4.4

;

[41*''"'-'' .rrr,l ,Pt,9I ;p2,q2;p3,q3; | .1*.2 , O.

OO

f

(d.,

+dr-1)

,' -.',dr+dr-r

)

du

zF

,Q2,Q3 ,X,Y,Z

IF (C1+C2) 8,8 ,9

CALL F(Cl,C2,P1, P2, P3,Q1,Q2,Q3, X,Y, Z,Nl,S) WRITE (n,5) CL ,C2,Pl, P2, P3,Q1 ,Q2,Q3, X, Y, Z,l'1, S [email protected](13H PARAI.{ETEp5 =,11F5 .2/3H M=,12,3H F=,1PE14.6) GOTA

dfd2 , t. 1

READ

(m,6) Cl ,C2 ,D\ ,D2

,X

6 F0Rr{AT (sF4 . 2) 9 READ(m,7)fl 7 FoRMAT(r2) 5 8

CALL F (C1,C2, D1, D2,X,It,S) I{RITE (n,5) C I ,C2 ,Dl, D2, X,M. S FoRI{AT(13H PARAI'IETERS- =, 5F5

Gora I

.2/sH M=',I2,3H F=,lpE14.6)

STOP END

1

STOP END

SUBRoUTINE

Lb

z("r ,cr;cir+u,dr-u;x) iIa;;nrdG;ut

I

6 FoRMAT(11F4.2) 9 READ(m,7)M 7 FoRMAT(12)

@*_.**

/

(Q2nANz- I .o) ) * (Y/AN2)

END

IF(D1+D2-1.0)8,8,9

8

.o) ) " (83/ (Q3+AN3-i . 0) ) * (Z/AN3)

1. CONTINUE

,L

5

1

CONTINUE

T2=T2* (A2/

2

END

.3

o

T3=T3* (A5l ( P3+AN3-

3

figHlil" 8..1

26t

T1=1.0

cAtL F(c1,c2,Dr,D2,G1,G2,H1,H2, X,Y,l,I,S)

T1

for Barnes and Related Integrals

S=0. o

6 F0tu"tAT ( 10F4 . 2) ni:nn (m, z) rq 7 ToRMAT ( I 2)

GOTO

Programs

SUBRoUTINE F(C1,C2,

s=0, o F

(C1, C2,Pt,P 2, P3,Ql,Q2, Q3, X,Y, Z,M, S)

T=1.0 D0 i Nl=1,M

Dl,D2, X;irl,S)

Programs fcir Barnes and Related lntegrals

262 AN1

lB.4

E2=El+.5 g-5+T T*T* ( (C1+AN1 -1 . o) / (Er+ANl - 1 . o) ) * ( (C2+AN1 - I . o) I (E2+ANl -1 .o) )

8.4.6

,,4s

*t+l-,1;l;l[ h1

*h2

bt. , c, ;b

Z,.ZtXr)l I :hr+u ;trr-,, :

r1 2

du

r(hr+u) r(hr-u)

"':]

!,

1 READ(m,6)81,82,C1,C2,G,Hl,H2,X,Y lF(Hl+H2-1)8,8,9

CALL F{A,Bt,B2,B3,GL,G2,G3,Hl,H2,I13,X,y,

B

STOP

SUBROUTINE S=o. O

=,

9F 5 .2/3Lt M=

,I2,3H

F=,1PE14.6)

ANI=FLOAT(Nt) A1=A+AN1-I.O T2=TI

F(Bl, B2,Cl,C2,G,H1,H2,X,Y,t*{,S)

A2=A1+AN2-1.O T3=T 2 [email protected] 3 N3=l ,lr{ AN3= FLOAT (N3 )

Tt=l.O

D0 I N1=1,M ANl=FL0AT(ll!)

A3=A2+AN3-1.0

E1=HI+H2-2.O+ANl

S=S+T3

(l=g+trNt-I.O

T3=T3" (A3l (G3+AN3-

T2=Tt

3

D0 2 N2=1,M AN2=FLOAT(N2) G2=G1+AN2-1 S=S+T2

.O

2

2=T2* ( (82+61.12 - 1 . O) / E2) * ( (Cz*4p2 -

C(,NI'INUE

l * ( (BI+AN1 - I

l{l:'t'Ut{N

liNll

'l

2 N2=1,M

AN2=FL0AT (N2)

O

.O

F(A,B1,B2,B5,Gl,C2,G3,HI,H2,H3,X,Y,Z,T{,S)

Tl=1.0 D0 I NI=1,lrl

[email protected]

E2=E1+AN2-I

Z,ll,s)

5

1

(:oN'l tNUn

,rr,l

(n, 5) A, 81,82,83, GI, G2,G3, H 1,H2,H3, X, y, Z,l\,1,S FpRI{AT(13H PARAMETERS =,13F 5.2/Ss M=,r2,3H F=,1P8I4.6) GOro 1

END

'l' l -.'f

g,h2;E,hr;

WRITE

8 STOP

T

+urhr -u;

;

END

[email protected](I3H PARAMETERS

S=0.

[ 'r,

b,I,LJ ; b, ; b"

6 FoRMAT(13F4.2) 9 READ(m,7j1,,1 7 FoRMAT(r2)

CALL F(B1,B2,Cl,C2,G,HI,H2, X,Y,ht,S) WRITE (n, 5)B1,B2,C1,C2,G,H1,H2, X,Y,l\{,S

[email protected]

)

,a:jr'

263

du -._o'81*hl' |(g'*u)r(h'-

6 FoRrrAT(9F4.2) 9 READ(m,7) 7 F0RI\,AT(12)

GOTO

I

for Barnes and Related Integrals

1 READ(m,6)A,B1,82,B3,Gl,G2,GS,H1,H2,H3,X,\,2 IF (G1+H1 - 1 .0) 8,8,9

-1

I

-+281+h,-t

CONTINUE RETURN

2

r(g,+h.,-1) |

T=T* (X/ANl)

END

5

Programs

=FL0AT (Nl )

[].([1+D2-1.O)/2.O

I

B.4I

.

O)

1 . O)

/Et ) * ( (C t+aNl - I

. O)

/ G2) * {Y / AN2) * 2

/GL) * (X/AN1) *2

1

1 . Ol

)

* ( (B3+Al.t3- t . O) / (H3+AN3-

1 . O)

* ) (Z/ANJ)

[email protected]

T2=T2* (A2/ (G2+tN2-1.0))*( (B2+AN2-1.o) / (H2+AN2-t.o) ) *(Y/AN2) CONTINUE

Dl=(G1+H1-t.o)/2.O s1=(Gl +trr)/2.o T1 =T1 * (A1l (D1 +AN1 - 1 . O) ) " ( (Bt*4111 CONTINUE RETURN END

- 1 . 0l

/ (Er+ANi - I . O) ) * (x/ANl)

i\ 264

Programs for Laplace lntegrals

lB.s

8.5]

8.5 Projrams for the Evaluation of lsplace Integrals

Prdgrams

[email protected]

S=0.

*,,u, Jo I e-Pt ta-I 2Fr(.t,.zidt,dr;xt) dt, d,P>O,

r

1 READ(m,6) A,Cl,C2,D1, IF(A)8,8,9 6 FoRMAT(7F4.2) 9 READ(m,7)I'1 7 FflRl'tAT ( 12 )

A1=A+A.ril-1.0 Dl=D+AN1-1.0

T2=Ti D0 2 N2=1

X

A2=A1+AN2-1.0 D2=Di+AN2-I.O S=S+T2

T2=T2* {AZ / P) * ( (82+AI,lz - | . o) / D2) * ( (C2+AN2 - 1 . o) T2=T2* (Y/AN2)

(n,5)A,C1,C2,Dl,D2,P,X,M,S) FORMAT(13H PAMMETERS =, 7F 5 .2/StI

WRITE

GOTL

U=

I

2

,I2,3H F=,IPE14.6)

8 STOP END

S=S+T T=T* ( (A+AN)

at

/P) * ( (C1 +411 / (Dl +AN) ) * ( (C2+AN) /

(D2+AN) )

t (x/

"-Pt

U

'

a-l

T_

O:2:21

o1

a,P>0,

'r

,br,.1ib2,c2:.

irl, ,

81 i

-P< x,I <

1 READ(m,6)A, B1, B2,Cl,C2,D,G1,G2, rF(A)8,8,9

EZ;

,.'''l

dt'

P.

P, X,Y

ronru,rtrrr+.2) READ(m,7)I'l FoRlrAT ( I2 )

)

(n, 5) A, Bl, 82,Cl,C2,D,CI,.G2,P,X, Y, M, S

5 lr0ltMAT(131{ PARAI'IETERS =, 11F5

60T0 | S1'@1'

,'-' ,rr(cl,dr;s'ht;xt)

U

(AN+1 .

*2F

o) )

6 9

z{.2,dr;E*hr;yt)

zGs,ds;Es,h-; > < a,P O, -P x+Y+7 < P. zF

zt)

dt

,

'

IF(A)8,8,9

FoRMAT(1 READ

N4

:ffH[f]r,ar,au,Dr,D2,D3,cr,G2,G3,Hl,H2,Hs,p,\,y,

z,r,{,s) CI, C2, C3, Dl,D2,D3, Gl,G2, G3,H1,H2,H3, P, X,Y, Z, Ivl,S [email protected] 1 1 3H PARAMETEB5 =, 9F 5. 2 / 2X,8F 5 . 2 /3H l,l=, 12, 3H F=, IPE14. 6)

$IRITE

5

7F4.2)

(m,7)

,3ffi9

(n,

5) A,

1

END

SUBROUTINE

CALL F(A, 81,82,Cl,C2,D,G1,q2,P,X,Y,M,S) WRITE

IINI)

(G1+AN I - 1 . o)

I READ(m,6)a,Cl,C2,C),Di,Dl,Dr,CI,G2,G3 ;t11,H2,H3,P,X,Y,2 @

8.s.2 =EIta) j|

I

e-Pt

CONTINUE

At

7

/

CONTINUE

T1=T1 * (A1/P) * ( (Bi +ANI - 1 . 0) /D1 ) * ( (C t +41q1 - 1 . o) T1=T1 " (X/AN1)

B.s.s tbf l,

RETURN END

6 9

(G2+AN2 -

.0

lfl=l,i;+iil, , . 1

I . 0) )

/

RETURN END

siJBRoUTtNE F(A,C1,C2,D|,D2.,P,X,i'J,S) S=0. O T=1

,M

AN2=FLOAT(N2)

CALL F(A,C1,C2,Dl,D2,P,X,T{,S)

5

F(A,B1,B2,Cl,C2,D,Gl,G2,P,X,Y,lu{,S)

O

ANI =FL0Ar(NI )

-P

1 READ(m,6) CL,C2,C3,C4,P,X1,X2,X3,

0,

cr+cr+ca+c4-P < 4'

X4

(P) 8,8, s FoRMAT(9F4.2) READ(n,7) l'{ rF

6 9

RETURN END

zPr(u,)r(c-p) r- , rz\ 8.6.3 mTtOTiE;:tf .} - o)"r (b1,b,br;c;-xt,-vt,-zt) dt' 5 " '0-tn-' Ocp

o.

READ(m,6)A, B,G,H; p, X,y

IF(A)8,8,9

6 FflRMAT ( 7F4 . 2) 9 READ(m,7)lvl 7 FoRrrAT(12) CALL F(A,B,G,H,P, X,Y,lU,S)

END

suBRot-[INE F(A,B,G1,G2,G3,1),X,Y,Z,M,S) O

I

5 8

T1=1.0

D0

I J

2)

FORMAT(I2)

S=0.

p

0

CALL F(A, B,G1,G2,G3,U,X,Y,Z,N!,S) WRITE

l-a-b Ifa+bl rta)r(b) " n!/l

NI=1,NI

/rl.lI =FL0AT (N1) P= (A+B+1 .O) /2.O E= (A-B+1

STOP END

D0

T2=Tt

T3.T2

M=,I2,3H F=,1pE14.6)

t

F

(A, B, G,H, P, X,Y,M,S)

T=1.0

EI=E+ANI-I.0

D2=D1+AN2-1.0 E2=El+AN2- I . O

G0r0

SUBROUTINE

Dl=D+ANI-1.0

D0 2 N2=1,M

, 5 J A, B, G, H, P , X, Y, M, S FORMAT(13H PARAMETERS =, 7FS.Z/SU

S=O. o

.o) / z.o

AN2=FLOAT (N2)

I{RITE (n

' I

I tl=r.u

AN=FLOAT(N) S=S+T

-1.o

T=T* ( (G+H+IN)

/

(A+B+AN) ) * (P/ (AN+r . O) ) * ( X+y)

CONTINUE

ffi;'* --a

h-

Programs for Convolution Integrals

L82

lB.e

283

Programs for Convolution Integrals

B.el 6 FoRMAT(r1F4.2J 9 READ (m, 7) M 7

va-L.(p-v.)-b-I

8.s.2 nr-a-bilffihT 0

xrFr(a+b;c; [x-)iv+yp)dv, a,b > o.

F0RI\IAT(Ii) CALL F(A,B,C,D,G1,G2,Hl,H2,P,-X,Y,1.1,S) WRITE (m,5)A, B,C, D,G7,G2,H1,H2rP, X,Y,I:,S [email protected](13H PARAMETERS =, 11F5 .2/3H tt=,12,3H F=, lPE14.6)

GLro t STOP END

I

READ (m, 6) A, B,C , P, X, Y

SUBROUTINE

I F (A) B, 8,9 FORMAT(6F4.2)

S=0.

6 9 READ(m,7)11 7 FoR-[lAT(I2)

Ti=1.0 D0 I N1=1,M AN1=FLOAT(N1)

CALL F[A, B,C, P, X,Y,I'{,S) WR

5 8

ITE ( n

,5)

, X, Y , I\{, PARAMETERS

A,

F0RMAi(13H

B,C,P

Al=A+B+AN1-1.0

S

=, 6F5'2/sH v=,12,3H

F=,1PE14'6)

CI=C+ANI-1.0 Dl=D+ANI-1.0

GATO I

T2=T1

STOP END

D0 2

A2=AI+AN2-I.O C2=C1+AN2-l.O

S=o.0

D2=D1+AN2-I

T1=1.O

I

T2=T2* (C2/ A2)* {D2/ (G2+AN2-1.0) ) * ( (B+AN2-1. 0) T2=T2* (Y/AN2) *P

Ct=C+AN1-1.0

CONTINUE

T2--T1

Tt=T1 * (C1 /A1) * (DI

D0 2 N2-1,M

lrll2=FL0AT(N2)

Tl=Ti*(X/ANl)*P

C2=CI+AN2-I.0

CONI'INUE

s=s+T2

T2=T2* ( (B+AN2- I .O)

I

.O

S=S+T2

Nl=,l,I4

ANl=FLOAT(N1)

2

N2=1 ,M

AN2=FL0AT (N2)

SUBROUTINE F (A, B,C, P, X,Y,I'I,S)

D0

F(A,B,C, D,CI,G2,HI,H2,P,X,Y,T,I,S)

O

CONTINUE Ti=Tr* ( (A+AN1-1

/

*P

o1-a-bi#h

8.s.4

CONTINUE RETURN END

READ

I . 0) )

P

.Jr'-t

(r-r)o-1rr,

(e1

;h, ;xv)

t*r, rp-rtvf u,o',|r|,l"_". . :91'ht;Ez,hz;93,h3; " L l a,b >

O.

tlliAlt(m, 6)A, B,C, D, G1,G2,Hl,H2,P, X,Y

,8,9

(rn,6)A, B, G1, G2, G3,Hl,H2,H3,?,X,Y,Z

IF(A)8,8,9

oi--T

_Lf

[H I+AN1 -

" rFr(82;hr;rh-vJ)rFr(sa;ha;z[p-v])dv, a,b > o.

B.s.s o1-a-b ri#B J,'-'(n-,)o-t

I t, (A) 8

/

0

p

I

(GI +ANi - 1 . o) ) * ( (A+AN I - I . o)

)

RETURN END

C2) * (Y/AN2) *P

.o)/c1) * (x/ANl)

/

I (H2+AN2-Lo)

ar,

6 FoRMAT(12F4.2) 9 READ (m, z) l"l '7

FORTIAT (

I2)

(A, B, GI, G2, G.3,H1,H2,H3, P, X,Y, Z,N{, S)WRITE (n, 5) A, B, G1, G2, G3,H1,H2,H3,P,X, Y, Z,Nt,S [email protected](13H PARAMETERS = ,12F 5.2/3H l'l= ,12,3H F=, lPE14.6) COTA T

CALL

F

lB.e

Prograrhs for Convolution lntegrals

284

8 STOP END I

SUBROffiINI F(A,ll,G1,G2,G3,HI,H2,H3'

t:

S'0.

Selected Bibliography

P' X'Y' Z'M's)

O

'l'I"l.O D0 I Nl=l

Abbott, l{.R. (1949)- Evaluation of an integral of a Bessel Function. J. l.\aLh. Physics 28 192-194. Abralrmanov, l!.A. and Abdikerimov, I.A. (1974). A certain Rienannliellin integral . Jsu. Akad. Ncuk. Razah S.S.fr. Ser. Fiz.-llat. 89 1-5 Abramorvitz, l{. et a}. (1965) . Handb,ook oi ltathematical Functions.

'lil ANI 'FLoAT (Nl )

Al'A+B+ANl-1,0

T2.Tl D0 2 N2=l,l'l AN2=rLoAT (N2)

A2'Al+AN2-1.0

Dover, llew York.

B2=B+AN2- I .0

Afshar, R. and l{ueller; F.14. (1975}. Hilbert tr:ansformation of densities of states using Hermite functions. J.Cornputational Phys. ll 190-209. Agahanov, S.A. and Natanson, G.I. (f968). The Lebesque function of Fourier-Jacobi sums. Vestnik Leningrad Uniu. 23 17-23. Agarwal , R.P. ( 196-
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