September 22, 2022 | Author: Anonymous | Category: N/A
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Èfk`ic 5. [che
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>.5. Jnbct`vj Lcfcrem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . >.>. Jnbct`vj Cspcièſtijs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9. @ftrjkuii`ùf
>
;. Kcserrjmmj [cùr`ij
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;.5. Tuftjs s`flumercs e`smekjs . . . . . . . . . . . . . . . . . . . ;.5.5. Tjmj kc jrkcf h . . . . . . . . . . . . . . . . . . . . . ;.5.>. Tuftj s`flumer cscfi`em . . . . . . . . . . . . . . . . ;.5.9. Tuftj s`flumer rchjv`nmc . . . . . . . . . . . . . . . ;.>. ^cs`kuj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.9. [cjrche kc Ieuiay . . . . . . . . . . . . . . . . . . . . . . . ;. ;.9. 9.5. 5. Iåmi Iåmium umj j kc rcs`k cs`kuj ujss hck` hck`ef eftc tc or orei eii` i`jf jfcs cs per peri` i`em emcs cs ;.;. ^cs`kujs y pjmjs . . . . . . . . . . . . . . . . . . . . . . . . . ;.;.5. Iåmiumj kcm ^cs`kuj cf uf Tjmj . . . . . . . . . . . . ;.;.>. ^cs`kujs em `fſtf`tj . . . . . . . . . . . . . . . . . . . ;.3. Epm`iei`ùf kc mjs rcs`kujs . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
> ; ? < 1 54 55 55 5> 5> 59
;.3.5. Iemiumj kc `ftclremcs `hprjp`es . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3. ^cocrcfi`e N`nm`jlråſties
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51
5
5. [che [cjrche kc rcs`kujs y pjmjs
>. Jnbct nbct`v `vjs js >. >.5. 5. Jnbc Jnbct` t`vj vj Lcfc Lcfcre remm Ijhprcfkcr Ijhpr cfkcr cm ijficptj y tcjrche kc rcs`kujs y pjmjs.
>.>.. Jnbct >.> Jnbct`v `vj j Cspciè Cspcièſtij ſtijss • Ijhprcfkcr cm tcjrche kc rcs`kujs y pjmjs. • Kcserrjmmer mes kchjstrei`jfcs y tcjrches.
9. @ftrj ftrjku kuii ii`ù `ùf f Cm tcjrche kc Ieuiay-Ljurset, Ieuiay-Ljurset, cstenmcic quc s` ufe oufi`ùf o cs efemèt`ie cf tjkjs mjs mjs puftjs `ftcr`jrcs e uf ijftjrfj icrrekj s`hpmc I y y sjnrc àm, cftjfics cm vemjr kc me `ftclrem kc k`iae oufi`ùf o cs icrj emrckckjr kc k`iaj ijftjrfj ijftjrfj I , e sencr2 Y`f chnerlj, chnerlj, s` me oufi`ùf oemme cf scr efemèt`ie pere uf fòhcrj ſtf`tj kc puftjs `ftcr`jrcs e I vcrchjs e ijft`fuei`ùf quc cx`stc uf fòhcrj cspcièſtij mmehekj rcs`kuj, pere ieke ufj kc mjs puftjs kjfkc o fj cs efemèt`ie, quc ijftr`nuyc em vemjr kc me `ftclrem. Esè quc cf mj quc s`luc kcserrjmmerchjs kcserrjmmerchjs me mmeheke tcjrèe kc mjs rcs`kujs, hck`eftc cm cfufi`ekj kc emlufjs tcjrches quc fjs scråf kc v`tem `hpjrtefi`e, cftrc cmmjs, cm tcjrche kc Ieuiay pere cm rcs`kuj.
;. Kcse Kcserr rrjm jmmj mj [cù cùr`i r`ij j ;.5.. Tuftj ;.5 Tuftjss s`flum s`flumer ercs cs e`smek e`smekjs js Ter ere e kcſtf kcſtf`r `r quà quà cs uf puft puftj j s` s`fl flum umer er e` e`sm smek ekj j kcnc kcnchj hjss rc rcij ijrk rker er quc quc uf puft puftj j z 4 sc mmeh mmehe e pufpuftj s`flumer kc ufe oufi`ùf o s` s` me oufi`ùf o cs cs fj efemèt`ie cf z 4 pcrj sè mj cs cf me vci`fkek kc z 4 .
Kcſtf`i`ùf 5. Sf puftj z 4 sc k`ic quc cs uf puftj s`flumer e`smekj kc ufe oufi`ùf o (z ), ), s` cstc cs s`flumer y, ekchås, cx`stc ufe vci`fkek kc z 4 cf tjkj puftj kc me iuem o ( z ) cs efemèt`ie cxicptj cf csc puftj. Cs kci`r, z 4 cs uf puftj s`flumer e`smekj s` o ( z ) cs efemèt`ie cf ufe vci`fkek 4 = |z ∞ z 4 | = γ. W5X
Cbchpmjs 5. Cfijftrer mjs puftjs s`flumercs e`smekjs kc me oufi`ùf o (z ) kcſtf`ke pjr o ( z ) 8
s`f z
z ∞ ;` Yjmui`ùf2 Yc t`cfc quc z 8 ;` cs cs cm òf`ij puftj s`flumer kc me oufi`ùf o(z), kekj quc o ( z ) cs cm ij o > (z ) 8 z ∞ ;` , s` i`cftc i`c ftc kc mes oufi`j oufi`jfcs fcs cftcr cftcres es o 5 (z ) 8 s`f z y y o s`cf cfkj kj ;` cm cm òf`i òf`ij j ic icrj rj kc o > (z ). ) . Esè Esè qu quc c
>
z 8 ;` cs uf puftj s`flumer e`smekj.
>. Cfijftrer mjs puftjs s`flumercs e`smekjs kc me oufi`ùf o (z ) keke pjr o ( z ) 8
z + 5 z (z > + 5)
z 9 8 ∞` . Yjmui`ùf2 Cf cstc iesj tcfchjs trcs puftjs s`flumercs e`smekjs, e sencr, z 5 8 4, z > 8 ` y z
Kekj quc o ( z ) cs cm iji`cftc kc mes oufi`jfcs cftcres o 5 (z ) 8 z + 5 y o > (z ) 8 z (z > + 5), s`cfkj 4 y s»` mjs òf`ijs icrj kc o > (z ). ). Esè quc cstjs trcs puftjs sjf s`flumercs e`smekjs. e`smekjs.
[` [`pjs pjs kc s`fl s`flume umer`ke r`kekcs kcs e`s e`smek mekes es Kcserrjmmj kc Meurcft me s`lu`cftc kcſtf`i`ùf etr`nu`kj e Meurcft, hucstre quc ufe oufi`ùf ijhj cste o ( z ) puckc cxpefk`rsc cf ufe scr`c kc pjtcfi`es pjs`t`ves y fclet`ves kc z ∞ z 4 . Cste scr`c cs me quc sc ijfjic ijhj scr`c kc Meurcft emrckckjr emrckckjr kc z 4 . W>X
Kcſtf`i Kcſt f`i`ùf `ùf >. Yce ∖
o ( z ) 8
f e f f (z ∞ z 4 ) +
f 84
∖
n ff
f f 85 (z ∞ z 4 )
cm kcserrjmmj kc Meurcft pere me oufi`ùf o ( z ) emrckckjr kcm puftj s`flumer e`smekj z 4 . Me pertc kcm kcserr kcs errjm jmmj mj kc Meurc Meurcft ft kc o ( z ) quc t`cfc t`cfc pjtcfi pjtcfi`es `es fclet` fclet`ves ves kc (z ∞ z 4 ) sc kcfjh kcfjh`fe `fe pertc pertc pr`fi` pr`fi`pem pem kc o cf z 4 . Cf jtres pemenres, cm tàrh`fj2 ∖ n f f f f 85 (z ∞ z 4 )
cs me pertc pr`fi`pem kc o cf z 4 .
Cbchpmj 5 cf ieke ufj kc sus puftjs s`flumercs. s`flumercs. z (z ∞ 5) Yjmui`ùf2 Cf cstc iesj, z > 8 5. iesj, mjs puftjs s`flumercs s`flumercs e`smekjs sjf z 5 8 4 y z Kctcrh`fc me pertc pr`fi`pem kc o ( z ) 8
`. Cm kcserrjmmj kcserrjmmj kc Meurcft pere o ( z ) emrckckjr kc z 5 cs
∖
5 z + o ( z ) 8 ∞ z f 84 f
kc kjfkc sc kckuic quc me pertc pr`fi`pem kc o ( z ) cf z 5 8 4 cs ∞
5 . z
``. Tjr jtre pertc, cm kcserrjmmj kc Meurcft pere o ( z ) emrckckjr kc z > cs ∖
o ( z ) 8
(∞5)f +5 (z ∞ 5)f +
f 84
5 z ∞ 5
kc kjfkc sc kckuic quc me pertc pr`fi`pem kc o ( z ) cf z > 8 5 cs
5 . z ∞ 5
Cf ehnjs iesjs sc ae ut`m`zekj cm kcserrjmmj kc Heimeur`f Heimeur`f 5 5 µ w
>
9
;
3
8 5 ∙ w + w ∙ w + w ∙ w + . . .
cm iuem cs våm`kj pere |w | = 5 9
;. ;.5. 5.5. 5. Tjmj jmj kc jr jrkc kcf fh Cf hetchåt`ies pjmjs y icrjs cs uf hàtjkj quc pcrh`tc cvemuer mjs pjmjs y mjs icrjs kc mes cxprcs`jfcs rei`jfemcs rei`jfemcs pere aemmer cm ijfbuftj sjmui`ùf cf kcs`luemkekcs. kcs`luemkekcs. Yu ut`m`kek rek`ie cf me lcfcrem`zei`ùf y hcief`zei`ùf kcm prjicsj. Cft`àfkesc pjr icrjs e mes cxprcs`jfcs pjm`fùh`ies quc ijfojrhef cm fuhcrekjr kc me oufi`ùf y pjr pjmjs e mes cxprcs`jfcs pjm`fùh`ies pjm`fùh`ies quc ijfojrhef cm kcfjh`fekjr iuefkj cstc t`cfkc e icrj, eprjx`håfkjsc me oufi`ùf cvemueke em `fſtf`tj.W9X
Kcſtf`i`ùf 9. Yce z 4 uf puftj s`flumer e`smekj kc ufe oufi`ùf o (z ). ). Yc k`ic quc z 4 cs uf pjmj kc jrkcf h kc kc o s` cm kcserrjmmj kc Meurcft pere pere o ( z ) tjhe me ojrhe ∖
o ( z ) 8
f e f f (z ∞ z 4 ) +
f ∞4
n 5
(z ∞ z 4 )
+
n >
(z ∞ z 4 )>
+...+
n h h
(z ∞ z 4 )h
kjfkc n h h +> 8 . . . 8 4 h +5 8 n h h 4 y n h Cf jtres pemenres, me pertc pr`fi`pem kc o cf z 4 pjscc cm ijcſti`cftc n h h 4 y mjs s`lu`cftcs ijcſti`cftcs sjf icrj. Mjs ijcſti`cftcs eftcr`jrcs e n h h fj fcicser`ehcftc sjf fumjs, pcrj puckcf scrmj.. Cf cm iesj cf quc h 8 5, z 4 sc kcfjh`fe pjmj s`hpmc. scrmj Me kcſtf`i`ùf eftcr`jr fjs `fk`ie quc pere kctcrh`fer s` z 4 cs uf pjmj kc o ( z ), ), prcv`ehcftc sc kcnc tcfcr cm kcserrjmmj kc Meurcft kc o ( z ) emrckckjr kc z 4 7 mj eftcr`jr fj cs pråit`ij, pcrj s`chprc cs våm`kj. Cx`stcf jtrjs prjick`h`cftjs hås ekciuekjs pere vcr`ſtier s` uf puftj cs j fj uf pjmj. Cm s`lu`cft lu` cftc c tcjrc tcjrche he fjs fjs hucst hucstre re uf prjic prjick`h k`h`cf `cftj tj pere pere vcr`ſti vcr`ſtier er s` uf puftj puftj cs uf pjmj pjmj s`f fcics` fcics`kek kek kc ijfstru`r su scr`c kc Meurcft.
[cjrche. Y` o (z ) t`cfc uf pjmj cf z 4 , cftjfics m» m»ĵh ĵhz ↞z 5 | o ( z )| 8 ∖ Kchjstrei`ùf. Yce o (z ) ufe oufi`ùf ijf uf pjmj kc jrkcf h cf z 4 , pjr mj quc cm kcserrjmmj kc Meurcft pere o ( z ) tjhe me ojrhe ∖
o ( z ) 8
f e f f (z ∞ z 4 ) +
f 84
n 5 n > n h h + +...+ > (z ∞ z 4 ) (z ∞ z 4 ) (z ∞ z 4 )h
kjfkc n h 4. Humt`pm`iefkj ehnjs mekjs pjr ( pjr ( z ∞ z 4 )h tcfchjs (z ∞ z 4 )h o ( z ) 8
∖
f +h h ∞5 h ∞> e f + n 5 (z ∞ z 4 ) + n > (z ∞ z 4 ) + . . . + n h f (z ∞ z 4 ) h
f 84
Kcm rcsumtekj eftcr`jr tcfchjs m»ĵh ĵh (z ∞ z 4 )h o ( z ) 8 n h h
z ↞z 4
ijhj me ciuei`ùf eftcr`jr `fk`ie quc
(z ∞ z 4 )h ↞ 4, iuefkj z ↞ z 4
m»ĵh ĵh | o ( z )| 8 ∖
z ↞z 4
ijf mj quc sc kchucstre cm cfufi`ekj kcm tcjrche. tcjrche. Cm tcjrche eftcr`jr fj sùmj fjs pcrh`tc `kcft`ſtier s` uf puftj cs uf pjmj, pjmj, s`fj tehn`àf cm jrkcf kcm h`shj. Nesåfkjsc cf cstc tcjrche, mes s`lu`cftcs rclmes fjs pcrh`tcf `kcft`ſtier cm jrkcf ;
kcm pjmj.
- ^clme clme @. Yce z 4 uf puftj s`flumer e`smekj kc o (z ). ). Y` cx`stc cm m»ĵh ĵh (z ∞ z 4 )h o ( z )
z ↞z 4
y s` k`iaj m`h`tc fj cs icrj f` `fſtf`tj, cftjfics o ( z ) t`cfc uf pjmj kc jrkcf h cf cf z 4 .
- ^c ^clm lmee @@. @@. Y` cm pjmj kc o (z ) cf z cs kc jrkcf h , cftjfics 4
m»ĵh ĵh (z ∞ z 4 )f o ( z ) 8
z ↞z 4
4 f 6 h ∖ f = h
Cbchpmj Cstenmczie cm jrkcf kc mjs pjmjs quc prcscfte me oufi`ùf 5 z > (z ∞ 9` )
o ( z ) 8
z 8 9` . Esè quc efem`ichjs cstjs Yjmui`ùf2 Cf cstc iesj vchjs quc o (z ) prcscfte pjmjs cf z 8 4 y z
pjmjs pjr scperekj.
`. Tere z 8 4 sc t`cfc quc
5 5 8 8∖ h 9 m»ĵh ĵh | o ( z )| 8 m»ĵĵh z ↞4 z ∞ 9` z > z ↞4 4 pjr mj quc cocit`vehcftc z 8 4 ijrrcspjfkc e uf pjmj kc o ( z ). ). _cehjs eajre eajre kc quà jrkcf cs cstc pjmj. Yce h uf uf cftcrj pjs`t`vj, esè quc
z h ∞>
m»ĵh ĵh z h o ( z ) 8 m»ĵh ĵh
z ↞4
z ∞ 9`
z ↞4
8
4 h 6 > ` 9
h 8 >
∖ h = >
Mj eftcr`jr s`lf`ſtie quc z 8 4 cs uf pjmj kc jrkcf > .
``. Tere z 8 9` sc sc t`cfc quc
5 5 8 8∖ h 9 m»ĵh ĵh | o ( z )| 8 m»ĵĵh z ↞9 z ↞9` z ∞ 9` z > 4 pjr mj quc cocit` cocit`veh vehcft cftc c z 8 9` ijrr ijrrcs cspj pjfk fkc c e uf pjmj pjmj kc o ( z ). ) . _ce cehj hjss eajr eajre e kc quà quà jrkc jrkcf f cs cs cstc tc pjmj. Yce h uf uf cftcrj pjs`t`vj, esè quc
(z ∞ 9` )h ∞5 h 8 m»ĵĵh h (z ∞ 9` ) o ( z ) 8 m»ĵĵh h z ↞9 z ↞9` z >
Mj eftcr`jr s`lf`ſtie quc z 8 9` cs cs uf pjmj kc jrkcf 5 .
3
4 h 6 5 ∞ 50 h 8 5 ∖ h = 5
Cf prjnmches rcmcveftcs cs ijhòf cfijftrersc ijf s`tuei`jfcs cf mes quc sc rcqu`crc kctcrh`fer cm jrkcf kc mjs pjmjs kc ufe oufi`ùf kc me ojrhe2 o ( z ) 8
p (z ) q (z )
Tjr mj quc e ijft`fuei`ùf cstenmcicrchjs cstenmcicrchjs uf tcjrche quc fjs pcrh`tc `kcft`ſtier s` uf puftj cs uf pjmj, pjmj, esè ijhj cm jrkcf kcm h`shj h`shj,, pere ufe oufi`ùf rei`jfem o ( z ). ).
[cjrche. Yce z 4 uf puftj kcm pmefj ijhpmcbj. Yce o (z ) ufe oufi`ùf tem quc o ( z ) 8
p (z ) q (z )
q (z ) sjf efemèt`ies cf z 4 y p (z 4 ) 4. Cftjfics, z 4 cs uf pjmj kc jrkcf h kc o , s` y kjfkc p (z ) y q sùmj s`, q (z 4 ) 8 q (z 4 ) 8 ¼ ¼ ¼ 8 q(h ∞5) (z 4 ) 8 4
y q (h ) (z 4 ) 4
Kchjstrei`ùf. Yupjflehjs quc p (z 4 ) 4, q (z 4 ) 8 q (z 4) 8 ¼ ¼ ¼ 8 q (h ∞5) (z 4 ) 8 4 y q q (h ) (z 4 ) 4. Y` p (z ) y q q (z ) sjf efemèt`ies cf z 4 ijf p (z 4 ) 4 y q q (z 4 ) 8 4, cftjfics z 4 cs uf puftj s`flumer q (z 4 ) 8 q (z 4 ) 8 ¼ ¼ ¼ 8 q (h ∞5) (z 4 ) 8 4 y e`smekj kc o ( ( z ). ). Tjr jtre pertc, ijhj q (z ) cs efemèt`ie cf z 4 y q q (h ) (z 4 ) 4, cftjfics cm kcserrjmmj kc [eymjr kc q (z ) emrckckjr kc z 4 cs ∖
q (z ) 8
f e f f (z ∞ z 4 )
f 8h
muclj
∖ q (z ) e ff (z ∞ z 4 )f ∞ht h + h 8 e h (z ∞ z 4 ) f 8h +5
pjr teftj, m»ĵh ĵh z ↞z 4
Kc cste ojrhe,
q (z )
8 e h h 4
(z ∞ z 4 )h
p (z ) m»ĵĵh h (z ∞ z 4 )h o ( z ) 8 m»ĵĵh h (z ∞ z 4 )h z ↞z 4 z ↞8 ↞84 q (z )
h
ĵhz ↞z 4 p (z ) m»ĵh ĵhz ↞z 4 (z ∞ z 4 ) o ( z ) 8 m»ĵh 5
8 p (z 4 ) e h h
4, ∖
z ∞z h m»ĵh ĵhz ↞z 4 ( q (z 4))
Mj iuem `fk`ie quc z 4 cs uf pjmj kc jrkcf h kc kc o ( z ))..
;.5.>. ;.5 .>. Tu Tuftj ftj s`fl s`flume umerr csc cscfi`e fi`emm Kcſtf`i Kcſt f`i`ùf `ùf ;. Yce z 4 uf puftj s`flumer e`smekj kc ufe oufi`ùf o ((z ). ). Yc k`ic quc z 4 cs uf puftj s`flumer cscfi`em, s` me pertc pr`fi`pem kc o cf z 4 t`cfc uf fòhcrj `fſtf`tj kc tàrh`fjs k`ocrcftcs kc icrj. W;X
?
Kc me kcſtf`i`ùf kc puftj s`flumer cscfi`em, sc kckuic quc z 4 cs uf puftj s`flumer cscfi`em kc o ( z ), ), s` y sùmj s`, m»ĵh ĵh o ( z ) z ↞z 4
fj cx`stc (f` ſtf`tj f` `fſtf`tj). Kc cste cste ojrh ojrhe, e, pe pere re kctc kctcrh rh`f `fer er s` uf puft puftj j s` s`fl flum umer er e`sm e`smek ekj j z 4 cs j fj uf pu puft ftj j s` s`fl flum umer er cs cscf cf-i`em kc o ( z ), ), fj cs fcicser`j ijfstru`r cm kcserrjmmj kc Meurcft kc o ( z ) emrckckjr kc z 4 .
Cbchpmj. 5/z Kchjstrer quc z 4 8 4 cs uf puftj s`flumer cscfi`em kc o ( z ) 8 c 5/z . Yjmui`ùf2 Kc eiucrkj em tcjrche eftcr`jr kcnchjs vcr`ſtier quc 5
m»ĵh ĵh c z
z ↞z 4
fj cx`stc. Tere cmmj, pjkchjs vcr quc s` fjs eicriehjs em jr`lcf pjr me rcite y 8 4, x 6 4, vchjs quc me oufi`ùf o ( z ) ircic s`f mèh`tc iuefkj x ↞ 47 h`cftres quc s` fjs eicriehjs em jr`lcf pjr me rcite y 8 4, x = 4, vchjs quc me oufi`ùf o ( z ) t`cfkc e icrj iuefkj x ↞ 4.
5/z 5/mèh`tc z mj eftcr`jr, eftcr`jr , pjkchjs quc sc s`flumer set`soeiccscfi`em cm tcjrche ye fj cx`stc, pjr mj qucIjf pjkchjs eſtrher quc z 4 8ijfimu`r 4 cs uf puftj kc o ( z quc ) 8 c cm .
Tjr jtrj mekj, mekj, s` fjs kci`k`hjs pjr ut`m`zer me kcſtf`i`ùf kc puftj s`flumer cscfi`em, cs fcicse5/z emrckckjr kc z 4 8 4 cstå kekj pjr r`j rcijrker quc cm kcserrjmmj kc Meurcft kc c 5/z 5/ 5/z z
c
∖
8 5+
5 f f 85 f !z
pere tjkj z tem tem quc |z | 6 4. Jnscrvehjs quc me pertc pr`fi`pem pjscc `fſtf`tjs ijcſti`cftcs k`st`ftjs kc icrj, pjr mj teftj, 5/z z 4 8 4 cs uf puftj s`flumer cscfi`em kc me oufi`ùf o ( z ) 8 c 5/z .
;.5.9. ;.5 .9. Tu Tuftj ftj s`fl s`flume umerr rchj rchjv`n v`nmc mc Kcſtf`i Kcſt f`i`ùf `ùf 3. Yce z 4 uf puftj s`flumer e`smekj kc ufe oufi`ùf o ((z ). ). Yc k`ic quc z 4 cs uf puftj s`flumer rchjv`nmc, rchjv`nmc, s` tjkjs mjs ijcſti`cftcs kc me pertc pr`fi`pem kc o cf z 4 sjf icrj. W;X Y` z 4 cs uf puftj s`flumer rchjv`nmc rchjv`nmc kc o ( z ), ), cm kcserrjmmj kc Meurcft tjhe me ojrhe ∖
o ( z ) 8
f e f f (z ∞ z 4 )
f 84
kc kjfkc sc kckuic quc m»ĵh ĵh o ( z ) 8 e 4
z ↞z 4
Tjr mj teftj, pere kctcrh`fer s` uf puftj s`flumer e`smekj z 4 cs uf puftj s`flumer rchjv`nmc kc o ( z ), ), neste ijf vcr`ſtier quc m»ĵh ĵh o ( z ) z ↞z 4
<
cx`stc y cs ſtf`tj.
Cbchpmj _ _cr`ſtier cr`ſtier quc z 4 8 4 cs uf puftj s`flumer rchjv`nmc rchjv`nmc kc o ( z ) 8 s`fz z . Yjmui`ùf2 Tere vcr`ſtier mj sjm`i`tekj nesterå hjstrer quc s`f z z ↞z 4 z m»ĵh ĵh
cx`stc7 esè quc epm`iefkj me rclme kc M‐Ajp`tem M‐Ajp`tem sc t`cfc quc s`f z ijs z 85 8 m»ĵh ĵh z ↞z 4 z z ↞z 4 5 m»ĵh ĵh
pjr mj teftj, em cx`st`r cm mèh`tc, sc kchucstre quc z 4 8 4 cs uf puftj s`flumer rchjv`nmc kc me oufi`ùf o ( z ) 8 s`fz z
;. ;.>. >. ^cs`k cs`kuj uj Kcſtf`i`ùf ?. Yce o ( z ) ufe oufi`ùf efemèt`ie sjnrc uf ijftjrfj icrrekj s`hpmc I y cf tjkj puftj `ftcr`jr e I , semvj cf z 4 . Cm rcs`kuj kc o ( z ) cf z 4 , quc sc kcfjte pjr ^csW o ( z ))XXz 8z 4 , cstå kcſtf`kj pjr 5 ^csW o ( z ))XXz ∞z 4 8 >τ` I o ( z )k z Iuefkj z 4 cs uf puftj s`flumer e`smekj kc ufe oufi`ùf o , aey uf fòhcrj pjs`t`vj ^ > tem quc o cs efemèt`ie cf tjkjs mjs puftjs z pere pere mjs iuemcs 4 = |z ∞ z 4 | = ^ > .W3X
Ijfsciucftchcftc, Ijfsciucf tchcftc, me oufi`ùf o ( z ) t`cfc ufe rcprcscftei`ùf rcprcscftei`ùf cf scr`c kc Meurcft ∖
o ( z ) 8
f
e f f (z ∞ z 4 ) +
f 84
∖
n ff
f f 85 (z ∞ z 4 )
kjfkc mjs ijcſti`cftcs e f f y n f f t`cfcf i`crte rcprcscftei`ùf `ftclrem. Cf pert`iumer, pert`iumer, mjs ijcſti`cftcs kc me pertc pr`fi`pem (n ff ) cståf kekjs pjr n f f 8 5 >τ`
o ( z )k∞z (f 8 5,>,...) f +5
I (z ∞ z 4 )
kjfkc I cs iuemqu`cr ijftjrfj s`hpmc icrrekj jr`cftekj pjs`t`vehcftc emrckckjr emrckckjr kc z 4 quc qucke kcftrj kcm k`sij pcrojrekj 4 = |z ∞ z 4 | = ^ > , tem ijhj sc hucstre cf me ſtlure.
O`lure 52 K`sij Tcrojrekj
1
Iuefkj f 8 5, me cxprcs`ùf pere n f f eftcr`jr sc rccsir`nc ijhj 5 n 5 8 >τ`
o ( z )k z
I
Y` sc ijhp ijhper ere e cste cste òm òmt` t`he he cx cxpr prcs cs`ù `ùf f ijf ijf me kcſtf kcſtf`i `i`ù `ùf f kc rc rcs` s`ku kuj j ufj ufj cfiu cfiucf cftr tre e quc quc cm fòhc fòhcrj rj ijhpmcbj n 5 , quc ijrrcspjfkc em ijcſti`cftc kc 5/ (z ∞ z 4 ) cf cm kcserrjmmj kc Meurcft pere o ( z ), ), cs prci`sehcftc prci`sehcf tc cm rcs`kuj kc o cf cm puftj s`flumer e`smekj e`smekj z 4 , pjr mj quc pjkchjs csir`n`r ^csW o ( z ))XXz 8z ee 8 n 5 Ijf tjkj mj eftcr`jr, pjkchjs ijfimu`r quc cm rcs`kuj kc me oufi`ùf o ( z ) cf cm puftj s`flumer e`smekj z 4 cs `luem em ijcſti`cftc kc ( kc ( z ∞ z 4 )∞5 cf me scr`c kc Meurcft quc rcprcscfte e o ( z ) cf ufe rcl`ùf efumer keke pjr 4 = |z ∞ z 4 | = ^ > , pere i`crtj fòhcrj rcem ^ > 6 4. Y` usehjs me kcſtf`i`ùf kc rcs`kuj, y cstc òmt`hj rcsumtekj, cfijftrehjs ufe cxprcs`ùf huy òt`m pere iemiumer i`crtes `ftclremcs emrckckjr kc ijftjrfjs icrrekjs s`hpmcs, e sencr
I
o ( z )k z 8 >τ` ^csW ^csW o ( z ))XXz 8z 4
s`chprc y iuefkj cstchjs cf ijfk`i`jfcs kc pjkcr cvemuer ^csW o ( z ))XXz 8z 4 .
Cbchpmj 5/z 5/ z
Iemiumc Iem iumc me `ftclrem `ftclrem I c tre cf me ſtlure efcxe.
k z kjfkc kjfkc I cs cm iè ièri rium umj j uf`t uf`ter er`j `j ic icft ftre rekj kj cf cm jr jr`l `lcf cf,, tem tem ijhj ijhj sc hucs hucs--
O`lure >2 I`riumj uf`ter`j icftrekj cf cm jr`lcf 5/z Yjmui`ùf2 Ijhj cm `ftclrefkj c 5/z cs efemèt`ij cf tjkj cm pmefj ijhpmcbj cxicptj cf cm jr`lcf, cftjfics t`cfc ufe rcprcscftei`ùf cf scr`cs kc Meurcft me iuem cs våm`ke pere tjkj z tem quc 4 = |z | = ∖7 pjr mj quc, cm vemjr kc me `ftclrem cs > τ` vcics vcics cm vemjr kcm rcs`kuj kc o ( z ) cf cm puftj z 8 4. 5/ z Tjr mj quc aerchjs usj kcm kcserrjmmj kc Meurcft pere me oufi`ùf o ( z ) 8 c 5/z emrckckjr kcm puftj z 8 4, cfijftrekj prcv`ehcftc, prcv`ehcftc, y quc cstå kekj pjr 5/z z c 5/ 8 5+
∖
5 f f 85 f !z
Kc cstc rcsumtekj, vchjs quc cm rcs`kuj kc o ( z ) cf cm puftj z 8 4 cs 5/z 5/z
^csW o ( z ))XXz 8z 4 8 ^cs c
pjr mj quc cs me sjmui`ùf
5/z z c 5/ k z 8 >τ`
I
0
z 84 8 5
;. ;.9. 9. [cjrc cjrche he kc Ieui Ieuiay ay Y` ufe oufi`ùf o cs cs efemèt`ie cf me rcl`ùf `ftcr`jr e uf ijftjrfj s`hpmc icrrekj I , cxicptj cf uf fòhcrj ſtf`tj kc puftjs, cstjs puftjs s`flumercs sjf e`smekjs. Cm s`lu`cftc tcjrche aeic usj cjrc rche he kc mjs mjs rcs` rcs`kuj kujss kc Ieuia Ieuiay y . W?X kc cstc rcsumtekj y sc ijfjic ijhj [cj
[cjrche. Yce I uf ijftjr ijftjrfj fj icrre icrrekj kj s`hpm s`hpmc c jr`cft jr`cftekj ekj pjs`t` pjs`t`veh vehcft cftc. c. Y` ufe oufi`ù oufi`ùf f o cs efeefe8 5,>,..., F ) `fmèt`ie kcftrj y sjnrc I , cxicptj cf uf fòhcrj ſtf`tj F kc puftjs s`flumercs z gg (g 8 tcr`jrcs e I , cftjfics
o ( z )k z 8 >τ`
I
F
^csW o ( z ))XXz 8z g g
g 85
Kchjstrei`ùf. Te Tere re prjner prjner cstc tcjrche tcjrche ijfs`kcr ijfs`kcrchjs chjs i`riufo i`riufocrc crcfi`es fi`es jr`cftekes jr`cftekes pjs`t`veh pjs`t`vehcftc cftc I g g , mj suſti`cftchcftc pcqucões pere quc fj tcflef puftjs cf ijhòf, icftrekes cf mes s`flumer`kekcs z g g c `ftcr`jrcs em ijftjrfj I , tem ijhj sc hucstre cf me ſtlure.
O`lure 92 Ijftjrfj icrrekj ijf F puftjs s`flumercs Mes i`riufocrcfi`es i`riufocrcfi`es I gg buftj ijf cm ijftjrfj I , ojrhef cm ijftjrfj kc ufe rcl`ùf icrreke cf me quc quc o cs efemèt efemèt`ie `ie y iuyj iuyj `ftcr` `ftcr`jr jr cs uf kjh`f` kjh`f`j j hòmt`p hòmt`pmch mchcft cftc c ijfcxj ijfcxj.. Esè quc, quc, ufe vcz e`smek e`smekes es mes s`flumer`kekcs, pjkchjs epm`ier cm [cjrche kc Ieuiay-Ljurset cm iuem cstenmcic quc "s` ufe oufi`ùf o cs cs efem efemèt èt`i `ie e cf tjkj tjkjss mjs mjs puft puftjs js `f `ftc tcr` r`jr jrcs cs e uf ijft ijftjr jrfj fj,, cftj cftjfi fics cs cm vemj vemjrr kc me `f `ftc tclr lrem em kc k`iae oufi`ùf o cs icrj emrckckjr emrckckjr kc k`iaj ijftjrfj", cf cstc iesj pjkchjs csir`n`r2
F
o ( z )k z ∞
I
o ( z )k z 8 4
g g 85 I g
quc sc rckuic e me cxprcs`ùf kcm tcjrche, tcjrche, ye quc
I gg
o ( z )k z 8 >τ` ^csW ^csW o ( z ))XXz 8z gg
ijf mj quc qucke kchjstrekj cm tcjrche kc mjs rcs`kujs kc Ieuiay.
Cbchpmj Iemiumer me `ftclrem z ∞ >
I
z (z ∞ 5) k z
kjfkc I cs me i`riufocrcfi`e |z | 8 9 jr`cfteke pjs`t`vehcftc. pjs`t`vehcftc.
54
Yjmui`ùf2 Yce o (z ) 8
z ∞> z (z ∞5) .
Mjs puftjs s`flumercs kc o sjf z 5 8 4 y z > 8 5. Yc jnscrve quc z > sjf pjmjs s`hpmcs kc o . Kc cste ojrhe, ehnjs sjf puftjs `ftcr`jrcs kc I , ekchås, z 5 y z
^csW o ( z ))XXz 8z 4 8 ^cs y ^csW o ( z ))XXz 8z 4 8 ^cs
z ∞ >
z (z ∞ 5)
z ∞ >
z (z ∞ 5)
8> z 84
8 ∞5
z 85
o cs ufe oufi`ùf efemèt`ie z > sjf puftjs s`flumercs e`smekjs kc o y o efemèt`ie cf I y y cf su `ftcr`jr, Ijhj z 5 y z z > , cftjfics pjr cm [cjrche kc mjs ^cs`kujs aemmehjs me sjmui`ùf. semvj cf z 5 y z
z ∞ >
I z (z ∞ 5)
k z 8 >τ` W> + (∞5)X 8 >τ`
;.9.5. ;.9 .5. Iåm Iåmium iumj j kc rcs`k rcs`kujs ujs hck hck`ef `eftc tc ore oreii`j ii`jfcs fcs peri`e peri`emcs mcs Cf s`tuei`jfcs cf mes quc tcfchjs ufe oufi`ùf rei`jfem o ( z ) kc me ojrhe o ( z ) 8
p (z ) q (z )
kjfkc q (z ) cs uf pjm`fjh`j kc lrekj heyjr em kc p (z ), ), cs pjs`nmc user me cxpefs`ùf cf oreii`jfcs peri`emcs pere cfijftrer mjs vemjrcs vemjrcs kcm rcs`kuj, rcs`kuj, tem ijhj mj cstenmcic cm s`lu`cftc tcjrche.
[cjrche. Yce o ( z ) ufe oufi`ùf rei`jfem, ijf F pjmjs s`hpmcs z gg , cftjfics me cxpefs`ùf cf oreii`jfcs peri`emcs kc o ( z ) t`cfc me ojrhe o ( z ) 8
E 5
(z ∞ z 5 )
+
E >
(z ∞ z > )
+
E 9
(z ∞ z 9 )
+ ... +
E F F
(z ∞ z F F )
kjfkc mjs fòhcrjs ijhpmcbjs E g , kcfjh`fekjs ijcſti`cftcs kc me cxpefs`ùf, cståf kekjs pjr
E gg 8 ^csW o ( z ))XXz 8z gg 8 (z ∞ z gg ) o ( z )
;. ;.;. ;. ^cs`k cs`kuj ujss y pjmj pjmjss
z 8z g g
Aeste equè, iuefkj ufe oufi`ùf o ( z ) t`cfc ufe s`flumer`kek cscfi`em cf z 4 , me òf`ie hefcre quc tcfchjs pere kctcrh`fer cm rcs`kuj cf k`iaj puftj ijfs`stc cf jntcfcr cm kcserrjmmj kc Meurcft emrckckjr kc z 4 y cmcl`r cmcl`r cm ijcſti`cftc eprjp`ekj, ijrrcspjfk`cftc em tàrh`fj ( tàrh`fj ( z ∞ z 4 )∞5 kc me pertc pr`fi`pem kc k`iaj kcserrjmmj. kcserrjmmj. Tcrj s` me oufi`ùf t`cfc uf pjmj cf z 4 fj cs fcicser`j jntcfcr tjkj cm kcserrjmmj kc Meurcft emrckckjr kc z 4 pere cfijftrer cm ijcſti`cftc quc nusiehjs. Cx`stcf k`vcrsjs hàtjkjs kc mjs quc pjkchjs ciaer hefj s`chprc y iuefkj scpehjs quc me s`fl s`flum umer er`k `kek ek cs uf pjmj pjmj,, ufj ufj kc cmmj cmmjss cs cm s` s`lu lu`c `cft ftc c tcjr tcjrch che e quc quc fjs fjs pc pcrh rh`t `tc c `k `kcf cft` t`ſti ſtier er s` uf pu pufftj z 4 cs uf pjmj kc o ( z ) y, ekchås, fjs k`ic iùhj iemiumer ^csW o ( z ))XXz 8z 4 .
55
;. ;.;. ;.5. 5. Iå Iåmi mium umj j kc kcmm ^cs`k cs`kuj uj cf uf Tjm jmj j [cjrche. Yce o (z ) ufe oufi`ùf kc ver`enmc ijhpmcbe. Yupjflehjs quc pere i`crtj cftcrj pjs`t`vj h , me oufi`ùf Ϗ(z ) 8 (z ∞ z 4 )h o ( z ) Ϗ (z 4 ) 4. Cftjfics o ( z ) t`cfc uf pjmj kc sc puckc kcſtf`r cf z 4 kc hjkj quc sce efemèt`ie eaè y Ϗ jrkcf h cf cf z 4 , y su su rcs`kuj eaè cstå kekj pjr Ϗ(h ∞5) (z 4 )
^csW o ( z ))XXz ∞z 4 8 (h ∞ 5)! _emc me pcfe fjter quc cf cm iesj cf quc cm pjmj cs s`hpmc (h 8 5), me cxprcs`ùf eftcr`jr sc rckuic _emc e ^csW o ( z )X )Xz 88 884 8 Ϗ (z 4 )
Kchjstrei`ùf. Ijf Ijfs`kcr s`kcrefkj efkj quc Ϗ(z ) cs efem efemèt èt`i `ie e cf z 4 , ekh ekh`tc `tc uf kcserr kcserrjm jmmj mj kc [ey eymjr mjr emrckc emrckckjr kjr kc z 4 , tem quc Ϗ(z ) 8 Ϗ (z 4 ) +
Ϗ (z 4 )
(z ∞ z 4 ) +
Ϗ (z 4 )
5! Ϗ (z 4 ) + (z ∞ z 4 )f f ! f 8h
>!
>
(z ∞ z 4 ) + . . . +
Ϗ(h ∞5) (z 4 )
(h ∞ 5)!
(z ∞ z 4 )(h ∞5)
(f )
∖
cf emlòf cftjrfj |z ∞ z 4 | = γ. Esè quc pjkchjs csir`n`r o ( z ) ijhj 5 Ϗ (z 4 ) Ϗ (z 4 ) Ϗ(h ∞5) (z 4 ) > o ( z ) 8 (z ∞ z 4 ) + (z ∞ z 4 ) + . . . + (z ∞ z 4 )(h ∞5) h Ϗ (z 4 ) + 5! >! (h ∞ 5)! (z ∞ z 4 ) ∖ Ϗ(f ) (z 4 ) + (z ∞ z 4 )f f ! f 8h
j
o ( z ) 8
Ϗ (z 4 )
Ϗ (z 4 )
+
(z ∞ z 4 )h
5! 5!((z ∞ z 4 )h ∞5 ∖ Ϗ(f ) (z 4 ) + (z ∞ z 4 )f ∞h f ! f 8h
+
Ϗ (z 4 )
>! (z ∞ z 4 )h ∞>
+ ... +
Ϗ(h ∞5) (z 4 )
(h ∞ 5)! 5)!((z ∞ z 4 )
iuefkj 4 = |z ∞ z 4 | = γ
Cstc kcserrjmmj cf scr`c kc Meurcft, buftj ijf cm aciaj kc quc Ϗ (z 4 ) 4, `hpm`ie quc - z 4 cs, cf cocitj, cocitj, uf pjmj kc jrkcf h kc kc o ( z ) - Cm ijcſti`cftc kc 5/ (z ∞ z 4 ) ijfſtrhe quc cm rcs`kuj kc o ( z ) cf z 4 cs Ϗ(h ∞5) (z 4 ) ^csW o ( z ))XXz ∞34 8 (h ∞ 5)! Ijf mj quc sc kchucstre cm tcjrche.
;.;. ;.;.>. >. ^c ^cs`k s`kuj ujss em `fſtf` `fſtf`tj tj [cjrche. Y` ufe oufi`ùf o cs efemèt`ie cf tjkj cm pmefj ſtf`tj e cxicpi`ùf cxicpi`ùf kc uf fòhcrj ſtf`tj kc puftj puftjss s`flum s`flumer ercs cs `ftcr` `ftcr`jr jrcs cs e uf ijftjr ijftjrfj fj icrre icrrekj kj s`hpmc s`hpmc I jr`cftekj jr`cftekj pjs`t`veh pjs`t`vehcftc cftc,, cftjfics cftjfics
o ( z )k z 8 >τ` ^cs ^cs
I
5>
5 5 o z z >
z 8z 4
;. ;.3. 3. Epm`i pm`iei ei`ù `ùf f kc mjs mjs rcs` rcs`ku kujs js ;.3.5. ;.3 .5. Ie Iemiu miumj mj kc `ftc `ftclre lremcs mcs `hpr `hprjp` jp`es es Cf iåmiumj, me `ftclrem `hprjp`e kc ufe oufi`ùf o ( x ) sjnrc me sch`rrcite sch`rrcite x ≣ 4 sc kcſtfc ijhj ^
∖
o ( x )k x 8 m»ĵh ĵh
o ( x )k x
^ ↞∖ ↞∖ 4
4
Iuefkj cm mèh`tc cx`stc sc k`ic quc me `ftclrem `hprjp`e cs ijfvcrlcftc y quc ijfvcrlc e k`iaj mèh`tc. Y` o ( x ) cs ijft`fue cf tjkj x , su `ftclrem `hprjp`e sjnrc me rcite ∞∖ = x = ∖ sc kcſtfc ijhj 4
∖
o ( x )k x 8 m»ĵh ĵh
^ 5 ↞∖ ∞^ 5
∞∖
o ( x )k x + m»ĵh ĵh
^ >
o ( x )k x
^ > ↞∖ 4
(5)
y iuefkj mjs kjs mèh`tcs cx`stcf sc k`ic quc me `ftclrem `hprjp`e ijfvcrlce me suhe kc ehnjs mèh`tcs.
Kcſtf`i`ùf )
↞∖ ∞^ ^ ↞∖
Cs `hpjrteftc hcfi`jfer quc s` me `ftclrem keke pjr me ciuei`ùf (>) cs ijfvcrlcftc, su vemjr pr`fi`pem kc Ieuiay (_T) cx`stc7 s`f chnerlj, me cx`stcfi`e kcm vemjr pr`fi`pem kc Ieuiay FJ lereft`ze me ijfvcrlcfi`e kc me `ftclrem (>), tem ijhj sc hucstre cf cm cbchpmj s`lu`cftc.
Cbchpmj Jnscrvchjs quc pere me `ftclrem
∖
xk x
∞∖
cm vemjr pr`fi`pem cs ∖
_ T
x k x 8 m»ĵh ĵh
∞∖
^
^ ↞∖ ↞∖ ∞^
xk x 8 m»ĵh ĵh
^ ↞∖ ↞∖
x >
^
>
8 m»ĵh ĵh W4X 8 4 ^ ↞∖ ↞∖
∞^
H`cftres quc usefkj me cxprcs`ùf (>) tcfchjs
∞∖
^ >
4
∖
x k x 8 m»ĵh ĵh
^ 5 ↞∖ ∞^ 5
8 m»ĵh ĵh
^ 5 ↞∖
x >
>
xk x + m»ĵh ĵh
^ > ↞∖ 4
4
+ m»ĵh ĵh
∞^ 5
^ 5>
8 ∞ m»ĵh ĵh
>
^ 5 ↞∖
^ > ↞∖
+ m»ĵh ĵh
^ > ↞∖
x >
>
xk x ^ > 4
^ >>
>
y kekj quc cstjs kjs òmt`hjs mèh`tcs fj cx`stcf, me `ftclrem `hprjp`e `hprjp`e fj cx`stc. ∖
xk x
∞∖
59
_ _emjr emjr pr`fi`pem pere ufe oufi`ùf `hper `hper.. Yupjflehjs quc o ( x ) cs ufe oufi`ùf `hper, `hper, tem quc pere tjkj x . o (∞x ) 8 ∞ o ( x )
Cf cstc iesj tcfchjs quc
^
o ( x )k x 8 4
∞^
ye quc cm årce nebj iurve kc o ( x ) un`ieke cftrc ∞^ y y 4 ieficme em årce nebj me iurve un`ieke cftrc 4 y ^ . Ijf mj eftcr`jr, cm vemjr pr`fi`pem kc Ieuiay pere ufe oufi`ùf `hper o ( x ) rcsumte ∖
_ T
o ( x )k x 8 m»ĵh ĵh
∞∖
^
o ( x )k x 8 4
^ ↞∖ ↞∖ ∞^
_ _emjr emjr pr`fi`pem pere ufe oufi`ùf per E ijft`fuei`ùf, supjflehjs supjflehjs quc o ( x ) cs ufe oufi`ùf per, per, tem quc o (∞x ) 8 o ( x )
pere tjkj x . Me s`hctrèe kc me oufi`ùf fjs pcrh`tc csir`n`r
^
^
o ( x )k x 8 >
∞^
o ( x )k x
4
mj quc pcrh`tc prjner quc pere ufe oufi`ùf per, cm vemjr pr`fi`pem kc Ieuiay cx`stc, y cstå cstå kekj pjr ∖
_ T
∞∖
o ( x )k x 8 >
∖
o ( x )k x
4
E ijft`fuei`ùf kcsir`n`rchjs uf hàtjkj nesekj cf mjs rcs`kujs quc pcrh`tc cvemuer `ftclremcs `hprjp`es kc oufi`jfcs rei`jfemcs percs o ( x ) kc me ojrhe2 o ( x ) 8
p (x ) q (x )
ijf o (∞x ) 8 o ( x ) q (x ) sjf pjm`fjh`js ijf ijcſti`cftcs rcemcs s`f oeitjrcs ijhufcs. y kjfkc p (x ) y q ijhufcs.
Yupjfchjs quc q (z ) fj t`cfc icrjs rcemcs y quc t`cfc em hcfjs uf icrj cf cm sch`pmefj ijhpmcbj supcr`jr (kjfkc @h( z ) 6 4 ). Cm hàtjkj sc `f`i`e `kcft`ſtiefkj tjkjs mjs icrjs k`st`ftjs kcm pjm`fjh`j q (z ) s`tuekjs cf cm sch`pm sch `pmefj efj supcr` supcr`jr jr,, aey uf fòhcr fòhcrj j ſtf`tj ſtf`tj kc cmmjs cmmjs (kcpcf (kcpcfk`c k`cfkj fkj kcm lrekj lrekj kcm pjm`f pjm`fjh jh`j) `j) y pjkcpjkchjs kcfjtermjs ijhj z 5 , z > , . . . , z f cs hcfjr j `luem quc cm lrekj F kc kc q (z ). ). f , kjfkc f cs E ijft`fuei`ùf sc `ftclre cm iji`cftc p (x ) o ( x ) 8 q (x ) 5;
O`lure ;2 Icrjs s`tuekjs cf cm sch`pmefj supcr`jr sjnrc cm ijftjrfj sch`i`riumer ojrhekj pjr cm sclhcftj rcitj kcskc z 8 ∞^ e z 8 ^ y me h`tek supcr`jr kc me i`riufocrcfi`e |z | 8 ^ kcsir`te kcsir`te cf scft`kj pjs`t`vj y quc quc kcfjterchjs pjr I ^^ , tem ijhj sc hucstre cf me ſtlure. Cf tjkj tjkj iesj iesj supjfk supjfkrc rchj hjss quc ^ cs cs mj suſti`c suſti`cftc ftchc hcftc ftc lrefk lrefkc c pere pere quc quc mjs mjs puftjs puftjs z 5 , z > , . . . , z ff scef tjkjs `ftcr`jrcs em ijftjrfj icrrekj quc sc cstå ijfs`kcrefkj. Ssefkj cm tcjrche kc mjs rcs`kujs kc Ieuiay y me perehctr`zei`ùf z 8 x pere cm sclhcftj rcitj,, pjkchjs csir`n`r2 rcitj
o ( z )k z 8
I
kc kjfkc
^
o ( x )k x +
∞^
o ( z )k z 8 >τ`
i ^ ^
o ( x )k x 8 >τ`
∞^
^csW o ( z ))XXz ∞z g g ∞
g 85
s`
m»ĵh ĵh
^ ↞∖ ↞∖ I ^ ^
cftjfics _ T
^csW o ( z ))XX8∞z gg
o ( z )k z
I ^ ^
o ( z )k z 8 4 f
^
f
g 85
f
^
o ( x )k x 8 >τ`
∞^
^csW o ( z ))XXz ∞z gg
g 85
[jhefkj cf iucfte quc o ( x ) cs ufe oufi`ùf per, per, pjkchjs csir`n`r ſtfemhcftc quc f
∖
o ( x )k x 8 >τ`
∞∖
^csW o ( z ))XXz 8z gg
g 85
y
f
∖
o ( x )k x 8 τ`
4
^csW o ( z ))XXz 8z g g
g 85
Cbchpmj ^csjmvcr me s`lu`cftc `ftclrem ∖
4
x > x ? + 5
k x
Tere cvemuer me `ftclrem ijhcfzehjs jnscrvefkj quc o ( z ) 8
z > z ? + 5
53
8
p (z ) q (z )
t`cfc s`flumer`kekcs e`smekes cf mjs icrjs kcm pjm`fjh`j q (z ) 8 z ? + 5, quc sjf mes reèics scxtes kc ∞5 8 c ` τ y cs efemèt`ie cf tjkjs mjs kchås puftjs kcm pmefj ijhpmcbj ijhpmcbj..
Cf cstc iesj, rcijrkefkj mj v`stj eftcr`jrhcftc pere cm iåmiumj kc reèics, vchjs quc
>g τ + z gg 8 i gg 8 cxp ` ? ? τ
ijf g 8 4,5,>,...,3 Mes pr`hcres pr`hcres trcs reèics reèics cståf cf cm sch`pmefj sch`pmefj supcr`jr y mes jtres trcs trcs cf cm sch`pmefj sch`pmefj `focr`jr, `focr`jr, tem ijhj sc hucstre cf me ſtlure.
O`lure 32 ^eèics cf cm sch`pmefj supcr`jr 4, 5,>) >) cståf cf cm `ftcr`jr kc me rcl`ùf sch`i`ri sch`i`riumer umer eijteke eijteke Iuefkj ^ 6 5, mjs puftjs i gg (g 8 4,5, pjr cm ijftjrfj icrrekj s`hpmc hjstrekj. Esè quc cf cstc iesj, `ftclrefkj sjnrc sjnrc cm ijftjrfj icrre icrrekj, kj, jntcfchjs jntcfchjs
o ( z )k z 8
I
^
o ( x )k x +
∞^
I ^ ^
o ( z )k z 8 >τ` WWN 4 + N 5 + N > X
kjfkc N g g (g 8 4,5,>). g cs cm rcs`kuj kc o ( z ) cf i g ), pjr Tere iemiumer mjs rcs`kujs eftcr`jrcs, vchjs quc mjs puftjs i gg sjf pjmjs s`hpmcs kc o ( z ), mj quc pjkchjs ut`m`zer uf tcjrche eftcr`jrhcftc v`stj quc cstenmcic quc ^cs
p (z ) q (z )
8 z 8z 4
p (z 4 ) q (z 4 )
Y` p (z 4 ) 4, q (z 4 ) 8 4 y q q (z 4 ) 4 Cf cstc iesj N gg 8 ^cs
z >
z ? µ 5
8 z 8i g g
> i g 3 ?i g
8
5 9 ?i g
3 ∞ >3 Tjr mj quc jntcfchjs ^
^ I ^
∞^
kc kjfkc rcsumte
5
o ( z )k z 8 >τ`
o ( x )k x +
^
o ( x )k x 8
∞^
τ
9
5?
∞
I ^ ^
?`
∞
5 ?`
o ( z )k z
+
5 ?`
8
τ
9
våm`ke pere tjke ^ , tem quc ^ 6 5. Cf cstc puftj tcfchjs quc ijfs`kcrer cm iåmiumj kc me `ftclrem kcm mekj kcrciaj, pere cmmj vehjs e ut`m`zer me perehctr`zei`ùf kc I ^ ^ keke pjr z (t ) 8 ^ c ` t
ijf4 ≫ t ≫ τ. E pert`r kc mj eftcr`jr, tcfchjs quc z > 8 |z |> 8 ^ >
y y,, usefkj me kcs`luemkek kcm tr`åflumj pere fòhcrjs ijhpmcbjs ( ijhpmcbjs ( |z 5 + z > | ≣ z 5 | ∞ |z > ), pjkchjs csir`n`r z ? + 5 ≣ ||z |? ∞ 5 |8 ^ ? ∞ 5 Esè quc s` z cs cs iuemqu`cr puftj sjnrc cm ijftjrfj I ^^ , cftjfics | o ( z )| 8
z > z ? + 5
^ >
≫
^ ? ∞ 5
Cf cstc puftj pjkchjs user uf rcsumtekj quc sc t`cfc iuefkj cvemuehjs ufe `ftclrem sjnrc uf ijftjrfj kc mjfl`tuk M cf cm quc cm hùkumj kc me oufi`ùf o ( z ) cstå eijtekj pjr uf vemjr H , cs kci`r
mj iuem `hpm`ie quc
I
o ( z )k z ≫
I ^ ^
cs kci`r
o ( z )k z ≫ H M
o ( z )k z
I ^ ^
^ >
^ ? ∞ 5
τ^ 8
τ^ 9
^ ? ∞ 5
τ ^ 9 8 ≫ ? 5 ^ 5 ∞ ^ 5? ∞ ? ^ ^ ? τ ^ 9
quc, cf cm mèh`tc iuefkj ^ ↞ ∖, rcsumte icrj.
Ijf cstc òmt`hj rcsumtekj pjkchjs cstenmcicr quc >
^
x k x 8 m»ĵĵh h ^ ↞∖ ↞∖ ∞^ x ? + 5
τ
9
kc kjfkc, cm vemjr pr`fi`pem kc Ieuiay cstå kekj pjr ∖
_ T
∞∖
x > x ? + 5
k x 8
τ
9
Mj quc pcrh`tc csir`n`r, csir`n`r, ſtfemhcftc, quc
∖
∞∖
x > x ? + 5
k x 8
τ
9
y ijhj cm `ftclrefkj cs per, per, tehn`àf tcfchjs cm rcsumtekj quc sc nusiene nusiene..
∖
4
x > x ? + 5
5<
k x 8
τ
?
3. ^cocr cocrcfi` cfi`ee N` N`nm nm`j `jlr lråſt åſtie iess W5X Kuertc ^. (>441). [cjrches kc Ieuiay y kcm ^cs`kuj. Hàx`ij2Sf`vcrs`kek Kc Yjfjre. ^ciupcrekj cm 5; kc scpt`chnrc kc >4>5, kc attps2//rpkuertc.ſts`ie.usjf.hx/eria`vjs/iursj9/43HctHetO`
[email protected] W>X Hurevia`g, I. A. (>44?). [refsojrheke R `fvcrse. ^csuhcf kc me tcjrĵe kc rcs`kujs pere me cvemuei`ù cvem uei`ùf f kc `ftclremc `ftclremcs. s. ^ciupcre ^ciupcrekj kj cm 5; kc scpt`chnr scpt`chnrc c kc >4>5, >4>5, kc attps2// attps2//ietck ietckre.`f re.`fl.uf l.ufmp mp.cku. .cku.er/ er/ cmcitrjtcif`e/scfys`s/ſtmc cmcitrjtcif `e/scfys`s/ſtmcs/epuftcs/^ s/epuftcs/^cpesj^cs`kujs.pko cpesj^cs`kujs.pko W9X Or Orefg efgs`f s`fg g J. (>45> (>45>). ). Tjmj Tjmjss y icrjs icrjs.. ^ciup ^ciupcre crekj kj cm 5; kc scpt`c scpt`chnr hnrc c kc >4>5, >4>5, kc attps2 attps2//c //cs. s.w`g w`g`vc `vcrr s`ty.jrl/w`g`/Tjmjsyicrjs W;X Hjmcrj H. (>444>5, >4>5, kc attp2//w attp2//www>. ww>.ieh` ieh`fjs fjs.uph .uph.cs/K .cs/Kcpert cpertehcf ehcftjs/ tjs/hetc hetchet` het`ies/ ies/ Ok`stefi`e/T@C/Efem`s`s>4hetchet`ij/[ches/I43^cs`kujs.pko W3X Krelef Krelef _. (>453). (>453). Kcperteh Kcpertehcftj cftj kc Hetchåt Hetchåt`ies `ies kc me SEH. SEH. @ftclre @ftclrei`ùf i`ùf ijhpmcb ijhpmcbe2 e2 [cjrche cjrche kc mj mjss ^cs`k ^cs`kujs ujs y sus epm`ie epm`iei`j i`jfcs fcs.. ^ciup ^ciupcr crekj ekj cm 5; kc scpt`c scpt`chn hnrc rc kc >4>5, >4>5, kc attp2/ attp2//vc /vcrsj rsj.he .het.u t.ue e h.cs/ krelef.vugjt`i/lrekj/ver-ijh-@-53-5?/verijh@->453-5?-fjtes-`ftclr-rcs.pko W?X Ljfzåm Ljfzåmcs cs E. (>440 (>440). ). Sf`vcrs f`vcrs`ke `kek k Ijhp Ijhpmut mutcfs cfsc c kc Hekr` Hekr`k. k. Hefue Hefuemm kc _er`enm er`enmc c Ijhpm Ijhpmcbe cbe.. ^c ^c-iupcrekj cm 5; kc scpt`chnr iupcrekj scpt`chnrc c kc >4>5, >4>5, kc attp2//be attp2//beijn` ijn`.ſts. .ſts.uih. uih.cs/e cs/ertch rtch`j/F `j/Fjtes jtes->4k ->4kc->4 c->4iurs iursj/_I j/_I.pko .pko W440) (>440).. Sf`vcrs f`vcrs`ke `kek k Tjm`t Tjm`tàif àif`ie `ie kc Ierte Iertelcf lcfe. e. Epuft Epuftcs cs kc _er`enm er`enmc c Ijhpm Ijhpmcbe cbe.. ^c^ciupcrekj cm 5; kc scpt`chnrc kc >4>5, kc attps2//www attps2//www.khec.upit.cs/ .khec.upit.cs/ bjsc/ehpiem/ijhpmcx.pko bjsc/ehpiem/ijhpmcx.pko
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