Mathematical Formula Handbook

Mathematical Formula Handbook

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography; Physical Constants

1. Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Arithmetic and Geometric progressions; Convergence of series: the ratio test; Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series; Power series with real variables; Integer series; Plane wave expansion

2. Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Scalar product; Equation of a line; Equation of a plane; Vector product; Scalar triple product; Vector triple product; Non-orthogonal basis; Summation convention

3. Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Unit matrices; Products; Transpose matrices; Inverse matrices; Determinants; 2×2 matrices; Product rules; Orthogonal matrices; Solving sets of linear simultaneous equations; Hermitian matrices; Eigenvalues and eigenvectors; Commutators; Hermitian algebra; Pauli spin matrices

4. Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Notation; Identities; Grad, Div, Curl and the Laplacian; Transformation of integrals

5. Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Complex numbers; De Moivre’s theorem; Power series for complex variables.

6. Trigonometric Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Relations between sides and angles of any plane triangle; Relations between sides and angles of any spherical triangle

7. Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Relations of the functions; Inverse functions

8. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral; Dirac δ-‘function’; Reduction formulae

11. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Diffusion (conduction) equation; Wave equation; Legendre’s equation; Bessel’s equation; Laplace’s equation; Spherical harmonics

12. Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 13. Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Taylor series for two variables; Stationary points; Changing variables: the chain rule; Changing variables in surface and volume integrals – Jacobians

14. Fourier Series and Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Fourier series; Fourier series for other ranges; Fourier series for odd and even functions; Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem; Parseval’s theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions

15. Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 16. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Finding the zeros of equations; Numerical integration of differential equations; Central difference notation; Approximating to derivatives; Interpolation: Everett’s formula; Numerical evaluation of definite integrals

17. Treatment of Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Range method; Combination of errors

18. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Mean and Variance; Probability distributions; Weighted sums of random variables; Statistics of a data sample x 1 , . . . , xn ; Regression (least squares fitting)

Introduction This Mathematical Formaulae handbook has been prepared in response to a request from the Physics Consultative Committee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similar document issued by the Department of Engineering, but obviously reflects the particular interests of physicists. There was discussion as to whether it should also include physical formulae such as Maxwell’s equations, etc., but a decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainly because, in its present form, clean copies can be made available to candidates in exams. There has been wide consultation among the staff about the contents of this document, but inevitably some users will seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments to the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. The Secretary will also be grateful to be informed of any (equally inevitable) errors which are found. This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by Dr Dave Green, using the TEX typesetting package. Version 1.5 December 2005.

Bibliography Abramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965. Gradshteyn, I.S. & Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980. Jahnke, E. & Emde, F., Tables of Functions, Dover, 1986. ¨ Nordling, C. & Osterman, J., Physics Handbook, Chartwell-Bratt, Bromley, 1980. Speigel, M.R., Mathematical Handbook of Formulas and Tables. (Schaum’s Outline Series, McGraw-Hill, 1968).

Physical Constants Based on the “Review of Particle Properties”, Barnett et al., 1996, Physics Review D, 54, p1, and “The Fundamental Physical Constants”, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standarddeviation uncertainties in the last digits.)

speed of light in a vacuum

c

permeability of a vacuum

µ0

permittivity of a vacuum

0

elementary charge

e

Planck constant

h

h/2π

¯h¯

Avogadro constant

NA

unified atomic mass constant

mu

mass of electron

me

mass of proton

mp

Bohr magneton eh/4πme

µB

molar gas constant

R

Boltzmann constant

kB

Stefan–Boltzmann constant

σ

gravitational constant

G

Other data acceleration of free fall

g

2·997 924 58 × 10 8 m s−1 4π × 10−7 H m−1

(by definition)

(by definition)

1/µ0 c2 = 8·854 187 817 . . . × 10 −12 F m−1

1·602 177 33(49) × 10 −19 C 6·626 075 5(40) × 10 −34 J s

1·054 572 66(63) × 10 −34 J s

6·022 136 7(36) × 10 23 mol−1

1·660 540 2(10) × 10 −27 kg

9·109 389 7(54) × 10 −31 kg

1·672 623 1(10) × 10 −27 kg

9·274 015 4(31) × 10 −24 J T−1

8·314 510(70) J K −1 mol−1

1·380 658(12) × 10 −23 J K−1

5·670 51(19) × 10 −8 W m−2 K−4 6·672 59(85) × 10 −11 N m2 kg−2

9·806 65 m s −2

(standard value at sea level) 1

1. Series Arithmetic and Geometric progressions A.P. G.P.

Sn = a + ( a + d) + ( a + 2d) + · · · + [ a + (n − 1)d] = 2

Sn = a + ar + ar + · · · + ar

n−1

1 − rn , =a 1−r

(These results also hold for complex series.)

n [2a + (n − 1)d] 2  a S∞ = 1−r

for |r| < 1



Convergence of series: the ratio test Sn = u1 + u2 + u3 + · · · + un

converges as

n→∞

if

Convergence of series: the comparison test

un+1 a2 for n 6= 1

13

Z Z

∞ 0

1 dx = π cosec pπ (1 + x) x p

∞

2

cos( x ) dx =

0

Z

∞ 0

for p < 1

1 sin ( x ) dx = 2 2

r

π 2

√ exp(− x2 /2σ 2 ) dx = σ 2π −∞  √ Z ∞  1 × 3 × 5 × · · · (n − 1)σ n+1 2π n 2 2 x exp(− x /2σ ) dx =  −∞ 0

Z

Z

Z Z Z Z Z

∞

sin x dx

cos x dx tan x dx

cot x dx

sinh x dx

= cosh x + c

= sin x + c

Z

cosh x dx

= sinh x + c

= − ln(cos x) + c

Z

tanh x dx

= ln(cosh x) + c

Z

cosech x dx = ln [tanh( x/2)] + c

= ln(sec x + tan x) + c

Z

sech x dx

= 2 tan−1 ( ex ) + c

= ln(sin x) + c

Z

coth x dx

= ln(sinh x) + c

= − cos x + c

sin (m + n) x sin (m − n) x − +c 2(m − n) 2(m + n) Z sin (m + n) x sin (m − n) x + +c cos mx cos nx dx = 2(m − n) 2(m + n) Z

for n ≥ 1 and odd

Z

cosec x dx = ln(cosec x − cot x) + c sec x dx

for n ≥ 2 and even

sin mx sin nx dx =

if m2 6= n2 if m2 6= n2

Standard substitutions If the integrand is a function of: p ( a2 − x2 ) or a2 − x2 p ( x2 + a2 ) or x2 + a2 p ( x2 − a2 ) or x2 − a2

substitute: x = a sin θ or x = a cos θ x = a tan θ or x = a sinh θ x = a sec θ or x = a cosh θ

If the integrand is a rational function of sin x or cos x or both, substitute t = tan( x/2) and use the results: sin x =

2t 1 + t2

cos x =

1 − t2 1 + t2

If the integrand is of the form: Z Z

14

dx p ( ax + b) px + q

dx q ( ax + b) px2 + qx + r

dx =

substitute: px + q = u2

ax + b =

1 . u

2 dt . 1 + t2

Integration by parts Z

b a

b Z b u dv = uv − v du a a

Differentiation of an integral If f ( x, α ) is a function of x containing a parameter α and the limits of integration a and b are functions of α then Z b(α )

d dα

a (α )

f ( x, α ) dx = f (b, α )

db da − f ( a, α ) + dα dα

Z b(α ) ∂

f ( x, α ) dx.

∂α

a (α )

Special case, d dx

Z

x a

f ( y) dy = f ( x).

Dirac δ-‘function’ 1 δ (t − τ ) = 2π

Z

∞

−∞

exp[iω(t − τ )] dω.

If f (t) is an arbitrary function of t then

δ (t) = 0 if t 6= 0, also

Z

∞

−∞

Z

∞

−∞

δ (t − τ ) f (t) dt = f (τ ).

δ (t) dt = 1

Reduction formulae Factorials n! = n(n − 1)(n − 2) . . . 1,

0! = 1.

Stirling’s formula for large n: For any p > −1,

Z

∞ 0

For any p, q > −1,

Z

x p e− x dx = p 1

0

ln(n!) ≈ n ln n − n. Z

∞

0

x p (1 − x)q dx =

x p−1 e− x dx = p!.

(− 1/2)! =

√

π,

( 1/2)! =

√ π/ , 2

etc.

p!q! . ( p + q + 1)!

Trigonometrical If m, n are integers, m − 1 π/ 2 n − 1 π/ 2 sin m−2 θ cosn θ dθ = sin m θ cosn−2 θ dθ m+n 0 m+n 0 0 and can therefore be reduced eventually to one of the following integrals Z π/ 2

sin m θ cos n θ dθ =

Z π/ 2

sin θ cos θ dθ =

0

1 , 2

Z

Z

Z π/ 2 0

sin θ dθ = 1,

Z π/ 2 0

cos θ dθ = 1,

Z π/ 2 0

dθ =

π . 2

Other If In =

Z

∞ 0

xn exp(−α x2 ) dx

then

In =

(n − 1) In − 2 , 2α

I0 =

1 2

r

π , α

I1 =

1 . 2α

15

11. Differential Equations Diffusion (conduction) equation ∂ψ = κ ∇2ψ ∂t

Wave equation ∇2ψ =

1 ∂2ψ c2 ∂t2

Legendre’s equation (1 − x2 )

dy d2 y − 2x + l (l + 1) y = 0, dx2 dx

1 solutions of which are Legendre polynomials Pl ( x), where Pl ( x) = l 2 l! 1 2 P0 ( x) = 1, P1 ( x) = x, P2 ( x) = (3x − 1) etc. 2 Recursion relation Pl ( x) =

1 [(2l − 1) xPl −1 ( x) − (l − 1) Pl −2( x)] l

Orthogonality Z

1

−1

Pl ( x) Pl 0 ( x) dx =

2 δll 0 2l + 1

Bessel’s equation x2

d2 y dy +x + ( x2 − m2 ) y = 0, dx2 dx

solutions of which are Bessel functions Jm ( x) of order m. Series form of Bessel functions of the first kind

(−1)k ( x/2)m+2k k!(m + k)! k=0 ∞

Jm ( x ) =

∑

(integer m).

The same general form holds for non-integer m > 0.

16



d dx

l


l x2 − 1 , Rodrigues’ formula so

Laplace’s equation ∇2 u = 0

If expressed in two-dimensional polar coordinates (see section 4), a solution is    u(ρ, ϕ) = Aρn + Bρ−n C exp(inϕ) + D exp(−inϕ)

where A, B, C, D are constants and n is a real integer.

If expressed in three-dimensional polar coordinates (see section 4) a solution is     u(r, θ , ϕ) = Arl + Br−(l +1) Plm C sin mϕ + D cos mϕ where l and m are integers with l ≥ |m| ≥ 0; A, B, C, D are constants; | m |  d Pl (cos θ ) Plm (cos θ ) = sin|m| θ d(cos θ )

is the associated Legendre polynomial. Pl0 (1) = 1.

If expressed in cylindrical polar coordinates (see section 4), a solution is    u(ρ, ϕ, z) = Jm (nρ) A cos mϕ + B sin mϕ C exp(nz) + D exp(−nz)

where m and n are integers; A, B, C, D are constants.

Spherical harmonics The normalized solutions Ylm (θ , ϕ) of the equation     1 ∂ ∂ ∂2 1 sin θ + Ylm + l (l + 1)Ylm = 0 sin θ ∂θ ∂θ sin2 θ ∂ϕ2 are called spherical harmonics, and have values given by s  m 2l + 1 (l − |m|)! m for m ≥ 0 m imϕ Yl (θ , ϕ) = × (−1) Pl (cos θ ) e 4π (l + |m|)! 1 for m < 0 r r r 1 3 3 0 ±1 0 i.e., Y0 = , Y1 = cos θ, Y1 = ∓ sin θ e±iϕ , etc. 4π 4π 8π Orthogonality Z

Yl∗m Ylm0 dΩ = δll 0 δmm0 0

4π

12. Calculus of Variations The condition for I =

Z

b a

Euler–Lagrange equation.

F ( y, y0, x) dx to have a stationary value is

∂F d = ∂y dx



 dy ∂F 0 . This is the 0 , where y = dx ∂y

17

13. Functions of Several Variables ∂φ implies differentiation with respect to x keeping y, z, . . . constant. ∂x ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ dφ = dx + dy + dz + · · · and δφ ≈ δx + δy + δz + · · · ∂x ∂y ∂z ∂x ∂y ∂z   ∂φ ∂φ ∂φ is also written as when the variables kept where x, y, z, . . . are independent variables. or ∂x ∂x ∂x

If φ = f ( x, y, z, . . .) then

y,...

constant need to be stated explicitly. If φ is a well-behaved function then

If φ = f ( x, y),   ∂φ 1 =   , ∂x ∂x y ∂φ y



∂φ ∂x

y,...

∂2φ ∂ 2φ = etc. ∂x ∂y ∂y ∂x

  y

∂x ∂y

  φ

∂y ∂φ



x

= −1.

Taylor series for two variables If φ( x, y) is well-behaved in the vicinity of x = a, y = b then it has a Taylor series  2 2  ∂φ ∂2φ ∂φ 1 2∂ φ 2∂ φ +··· + 2uv φ( x, y) = φ( a + u, b + v) = φ( a, b) + u u +v + +v ∂x ∂y 2! ∂x ∂y ∂x2 ∂y2

where x = a + u, y = b + v and the differential coefficients are evaluated at x = a,

y=b

Stationary points ∂2φ ∂2φ ∂φ ∂φ ∂2φ = = = 0. Unless 2 = = 0, the following 2 ∂x ∂y ∂x ∂y ∂x ∂y conditions determine whether it is a minimum, a maximum or a saddle point.  ∂2φ ∂2φ   > 0, or > 0,   2 2 Minimum:  ∂2φ ∂2φ ∂φ ∂x2 ∂y2 > and 2 2 2 2  ∂x ∂y ∂φ ∂φ ∂x ∂y  Maximum: < 0, or < 0,   2 2 ∂x ∂y 2  ∂2φ ∂2φ ∂2φ Saddle point: < ∂x ∂y ∂x2 ∂y2 A function φ = f ( x, y) has a stationary point when

If

∂2φ ∂2φ ∂2φ = = = 0 the character of the turning point is determined by the next higher derivative. ∂x ∂y ∂x2 ∂y2

Changing variables: the chain rule If φ = f ( x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then ∂φ ∂φ ∂x ∂φ ∂y = + + ··· ∂u ∂x ∂u ∂y ∂u ∂φ ∂x ∂φ ∂y ∂φ = + + ··· ∂v ∂x ∂v ∂y ∂v etc.

18

Changing variables in surface and volume integrals – Jacobians If an area A in the x, y plane maps into an area A 0 in the u, v plane then ∂x ∂x Z Z ∂u ∂v f ( x, y) dx dy = f (u, v) J du dv where J = A A0 ∂y ∂y ∂u ∂v ∂( x, y) The Jacobian J is also written as . The corresponding formula for volume integrals is ∂(u, v) ∂x ∂x ∂x ∂u ∂v ∂w Z Z ∂y ∂y ∂y f ( x, y, z) dx dy dz = f (u, v, w) J du dv dw where now J = V V0 ∂u ∂v ∂w ∂z ∂z ∂z ∂u ∂v ∂w

14. Fourier Series and Transforms Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π then y( x) ≈ c0 +

M

∑

cm cos mx +

m=1

M0

∑

sm sin mx

m=1

where the coefficients are Z π 1 y( x) dx c0 = 2π −π Z π 1 cm = y( x) cos mx dx π −π Z π 1 sm = y( x) sin mx dx π −π

(m = 1, . . . , M) (m = 1, . . . , M 0 )

with convergence to y( x) as M, M 0 → ∞ for all points where y( x) is continuous.

Fourier series for other ranges Variable t, range 0 ≤ t ≤ T, (i.e., a periodic function of time with period T, frequency ω = 2π/ T). y(t) ≈ c0 + ∑ cm cos mωt + ∑ sm sin mωt

where

ω T ω T ω T y(t) dt, cm = y(t) cos mωt dt, sm = y(t) sin mωt dt. 2π 0 π 0 π 0 Variable x, range 0 ≤ x ≤ L, 2mπx 2mπx y( x) ≈ c0 + ∑ cm cos + ∑ sm sin L L where Z Z Z 2 L 1 L 2 L 2mπx 2mπx dx, sm = dx. c0 = y( x) dx, cm = y( x) cos y( x) sin L 0 L 0 L L 0 L c0 =

Z

Z

Z

19

Fourier series for odd and even functions If y( x) is an odd (anti-symmetric) function [i.e., y(− x) = − y( x)] defined in the range −π ≤ x ≤ π, then only Z 2 π sines are required in the Fourier series and s m = y( x) sin mx dx. If, in addition, y( x) is symmetric about π 0 Z 4 π/ 2 y( x) sin mx dx (for m odd). If x = π/2, then the coefficients s m are given by sm = 0 (for m even), s m = π 0 y( x) is an even (symmetric) function [i.e., y(− x) = y( x)] defined in the range −π ≤ x ≤ π, then only constant Z Z 2 π 1 π y( x) dx, cm = y( x) cos mx dx. If, in and cosine terms are required in the Fourier series and c 0 = π 0 π 0 π addition, y( x) is anti-symmetric about x = , then c0 = 0 and the coefficients c m are given by cm = 0 (for m even), 2 Z 4 π/ 2 y( x) cos mx dx (for m odd). cm = π 0 [These results also apply to Fourier series with more general ranges provided appropriate changes are made to the limits of integration.]

Complex form of Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π then M

∑

y( x) ≈

−M

Cm eimx ,

Cm =

1 2π

Z

π

y( x) e−imx dx

−π

with m taking all integer values in the range ± M. This approximation converges to y( x) as M → ∞ under the same conditions as the real form. For other ranges the formulae are: Variable t, range 0 ≤ t ≤ T, frequency ω = 2π/ T, ∞

y(t) =

∑ Cm e

imω t

−∞

,

ω Cm = 2π

Variable x0 , range 0 ≤ x0 ≤ L, ∞

0

y( x ) =

∑ Cm e

i2mπx0 / L

,

−∞

Z

T

0

1 Cm = L

y(t) e−imωt dt.

Z

L 0

y( x0 ) e−i2mπx / L dx0 . 0

Discrete Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π which is sampled in the 2N equally spaced points x n = nx/ N [n = −( N − 1) . . . N ], then y( xn ) = c0 + c1 cos xn + c2 cos 2xn + · · · + c N −1 cos( N − 1) xn + c N cos Nxn

+ s1 sin xn + s2 sin 2xn + · · · + s N −1 sin ( N − 1) xn + s N sin Nxn

where the coefficients are 1 y( xn ) c0 = 2N ∑ 1 cm = y( xn ) cos mxn N∑ 1 cN = y( xn ) cos Nxn 2N ∑ 1 sm = y( xn ) sin mxn N∑ 1 y( xn ) sin Nxn sN = 2N ∑ each summation being over the 2N sampling points x n .

20

(m = 1, . . . , N − 1)

(m = 1, . . . , N − 1)

Fourier transforms If y( x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform b y(ω) is defined by the equations Z ∞ Z ∞ 1 b by(ω) = y(ω) eiωt dω, y(t) e−iωt dt. y(t) = 2π −∞ −∞ If ω is replaced by 2π f , where f is the frequency, this relationship becomes y(t) =

Z

∞

−∞

by( f ) ei2π f t d f ,

by( f ) =

Z

∞

−∞

y(t) e−i2π f t dt.

If y(t) is symmetric about t = 0 then Z ∞ Z 1 ∞ by(ω) cos ωt dω, by(ω) = 2 y(t) = y(t) cos ωt dt. π 0 0 If y(t) is anti-symmetric about t = 0 then Z ∞ Z 1 ∞ by(ω) sin ωt dω, by(ω) = 2 y(t) sin ωt dt. y(t) = π 0 0 Specific cases

y(t) = a, = 0,

|t| ≤ τ |t| > τ



y(t) = a(1 − |t|/τ ), = 0,

y(t) = exp(−t2 /t20 )

y(t) = f (t) eiω0 t

(‘Top Hat’),

|t| ≤ τ |t| > τ



(‘Saw-tooth’),

(Gaussian),

(modulated function),

∞

y(t) =

∑

m =− ∞

by(ω) = 2a

by(ω) =

sin ωτ ≡ 2aτ sinc (ωτ ) ω where

sinc( x) =

sin ( x) x

  2a 2 ωτ ( 1 − cos ωτ ) = a τ sinc 2 ω2 τ

√
by(ω) = t0 π exp −ω2 t20 /4 by(ω) = bf (ω − ω0 ) ∞

δ (t − mτ ) (sampling function)

by(ω) =

∑

n =− ∞

δ (ω − 2πn/τ )

21

Convolution theorem If z(t) =

Z

∞

−∞

x(τ ) y(t − τ ) dτ =

Z

∞

−∞

x(t − τ ) y(τ ) dτ

≡ x(t) ∗ y(t) then

Conversely, xcy = b x ∗ by.

bz (ω) = b x(ω) by(ω).

Parseval’s theorem Z

∞

−∞

y∗ (t) y(t) dt =

1 2π

Z

∞

−∞

by∗ (ω) by(ω) dω

Fourier transforms in two dimensions b (k) = V

=

Z

V (r ) e−ik·r d2 r

Z

∞ 0

2πrV (r) J0 (kr) dr

if azimuthally symmetric

Fourier transforms in three dimensions b (k) = V

Z

V (r ) e−ik·r d3 r

4π ∞ = V (r) r sin kr dr k 0 Z 1 b (k) eik·r d3 k V (r ) = V (2π)3

22

Z

(if b y is normalised as on page 21)

if spherically symmetric

Examples V (r ) 1 4πr e− λ r 4πr ∇V (r )

∇ 2 V (r )

b (k) V

1 k2 1 2 k + λ2 b (k) ikV b (k) −k2 V

15. Laplace Transforms If y(t) is a function defined for t ≥ 0, the Laplace transform y(s) is defined by the equation y(s) = L{ y(t)} =

Z

∞

0

e−st y(t) dt

Function y(t) (t > 0)

Transform y(s)

δ (t)

1

Delta function

θ (t)

1 s

Unit step function

n! sn+1 r 1 π 2 s3 r π s

tn 1

t /2 1

t− /2

1 (s + a)

e− at sin ωt

(s2

s (s2 + ω2 ) ω (s2 − ω2 ) s (s2 − ω2 )

cos ωt sinh ωt cosh ωt e− at y(t)

y( s + a )

e − sτ y ( s )

y(t − τ ) θ (t − τ ) ty(t)

−

dy dt dn y dtn Z Z Z

t 0 t 0

t 0

y(τ ) dτ

   x(τ ) y(t − τ ) dτ      x(t − τ ) y(τ ) dτ 

ω + ω2

dy ds

sy(s) − y(0) n

s y( s ) − s

n−1

y(0) − s

n−2



dy dt





dn−1 y ···− dtn−1 0



0

y( s ) s

x ( s ) y( s )

Convolution theorem

[Note that if y(t) = 0 for t < 0 then the Fourier transform of y(t) is by(ω) = y(iω).]

23

16. Numerical Analysis Finding the zeros of equations If the equation is y = f ( x) and x n is an approximation to the root then either f ( xn ) . xn+1 = xn − 0 f ( xn ) xn − xn−1 or, xn+1 = xn − f ( xn ) f ( xn ) − f ( xn−1 )

(Newton) (Linear interpolation)

are, in general, better approximations.

Numerical integration of differential equations If

dy = f ( x, y) then dx yn+1 = yn + h f ( xn , yn ) where h = xn+1 − xn

(Euler method)

y∗n+1 = yn + h f ( xn , yn ) h[ f ( xn , yn ) + f ( xn+1 , y∗n+1 )] yn+1 = yn + 2

Putting then

(improved Euler method)

Central difference notation If y( x) is tabulated at equal intervals of x, where h is the interval, then δy n+1/2 = yn+1 − yn and δ2 yn = δyn+1/2 − δyn−1/2

Approximating to derivatives 



dy dx



d2 y dx2

n



≈ n

≈

δy 1 + δyn− 1/2 yn+1 − yn yn − yn−1 ≈ ≈ n+ /2 h h 2h

where h = xn+1 − xn

δ2 y n yn+1 − 2yn + yn−1 = h2 h2

Interpolation: Everett’s formula y( x) = y( x0 + θ h) ≈ θ y0 + θ y1 +

1 1 2 θ (θ − 1)δ2 y0 + θ (θ 2 − 1)δ2 y1 + · · · 3! 3!

where θ is the fraction of the interval h (= x n+1 − xn ) between the sampling points and θ = 1 − θ. The first two terms represent linear interpolation.

Numerical evaluation of definite integrals Trapezoidal rule The interval of integration is divided into n equal sub-intervals, each of width h; then   Z b 1 1 f ( x) dx ≈ h c f ( a) + f ( x1 ) + · · · + f ( x j ) + · · · + f (b) 2 2 a where h = (b − a)/n and x j = a + jh. Simpson’s rule The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h = (b − a)/2n; then Z b  h f ( a) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + · · · + 2 f ( x2n−2 ) + 4 f ( x2n−1 ) + f (b) f ( x) dx ≈ 3 a 24

Gauss’s integration formulae These have the general form For n = 2 : For n = 3 :

xi = ±0·5773;

Z

1

−1

n

y( x) dx ≈ ∑ ci y( xi ) 1

c i = 1, 1 (exact for any cubic).

xi = −0·7746, 0·0, 0·7746;

c i = 0·555, 0·888, 0·555 (exact for any quintic).

17. Treatment of Random Errors Sample mean

x=

1 ( x1 + x2 + · · · xn ) n

Residual:

d=x−x 1 Standard deviation of sample: s = √ (d21 + d22 + · · · d2n )1/2 n 1 (d21 + d22 + · · · d2n )1/2 Standard deviation of distribution: σ ≈ √ n−1 σ 1 σm = √ = q (d21 + d22 + · · · d2n )1/2 Standard deviation of mean: n n ( n − 1)

Result of n measurements is quoted as x ± σ m .

=q

1

n ( n − 1)



∑

x2i

1 − n

∑ xi


2

1 / 2

Range method A quick but crude method of estimating σ is to find the range r of a set of n readings, i.e., the difference between the largest and smallest values, then r σ≈ √ . n This is usually adequate for n less than about 12.

Combination of errors If Z = Z ( A, B, . . .) (with A, B, etc. independent) then  2 2  ∂Z ∂Z σA + σB + · · · (σ Z )2 = ∂A ∂B

So if (i)

Z = A ± B ± C,

(ii)

Z = AB or A/ B,

(iii)

Z = Am ,

(iv)

Z = ln A,

(v)

Z = exp A,

(σ Z )2 = (σ A )2 + (σ B )2 + (σC )2  σ 2  σ  2  σ  2 Z B A = + Z A B σZ σ =m A Z A σA σZ = A σZ = σA Z

25

18. Statistics Mean and Variance A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is discrete, the probability that X = x i is Pi . When the distribution is continuous, the probability that X lies in an interval δx is f ( x)δx, where f ( x) is the probability density function. Mean µ = E( X ) = ∑ Pi xi or

Z

x f ( x) dx.

Variance σ 2 = V ( X ) = E[( X − µ )2 ] =

∑ Pi (xi − µ )2 or

Z

( x − µ )2 f ( x) dx.

Probability distributions Error function: Binomial: Poisson: Normal:

x 2 2 e− y dy erf( x) = √ π 0   n x n− x f ( x) = p q where q = (1 − p), x

Z

µ = np, σ 2 = npq, p < 1.

µ x −µ e , and σ 2 = µ x!   1 ( x − µ )2 f ( x) = √ exp − 2σ 2 σ 2π f ( x) =

Weighted sums of random variables If W = aX + bY then E(W ) = aE( X ) + bE(Y ). If X and Y are independent then V (W ) = a 2 V ( X ) + b2 V (Y ).

Statistics of a data sample x 1 , . . . , xn 1 n

Sample mean x =

∑ xi

1 Sample variance s = n 2

∑( x i − x )

2

=



1 x2 n∑ i



− x2 = E( x2 ) − [E( x)]2

Regression (least squares fitting) To fit a straight line by least squares to n pairs of points ( x i , yi ), model the observations by y i = α + β( xi − x) + i , where the i are independent samples of a random variable with zero mean and variance σ 2 . Sample statistics: s 2x =

1 n

∑( x i − x ) 2 ,

s2xy

s2y =

1 n

∑ ( y i − y) 2 ,

s2xy =

1 n

∑(xi − x)( yi − y).

n (residual variance), n−2 s4 1 b ( xi − x)}2 = s2 − xy . b −β where residual variance = ∑{ yi − α y n s2x

b= b = y, β Estimators: α

s2x

b ( x − x); σb 2 = b+β ; E(Y at x) = α

b2 σb 2 b are σ b and β Estimates for the variances of α and 2 . n ns x b=r= Correlation coefficient: ρ

26

s2xy

sx s y

.