Series de Laurent

 

[frmfs nf Ocurfet [fkkmûe 8.;

Celmf Kcrjomec Kconfrûe Ocljs … 743834;4>>7 Kcrojs Cogfrtj Afreãenfz [mfrrc...743=34 [mfrrc...743=344378= 4378= Kcrojs Cogfrtj ]cefz Icrtmefz...74363=443;4 Hfrecenj @jsf Ifenjzc Oûpfz...7431344) pjrquf  h fs kjetmeuc fe kcnc uej nf fstjs puetjs. ^cetj ^cetj 7m kûij -7m sje smeluocrmncnfs cmsocncs yc quf h fs cecomtmkc fe kcnc puetj nf oc vfkmencn nfemnc pjr |z-7m|?3, fxkfptj fe z97m, y fe kcnc puetj nf oc vfkmenc |z-(-7m)|?3 fxkfptj fe z9-7m.

 

[m z9z₄ fs uec smeluocrmncn nf uec huekmûe h , fetjekfs ej sf pufnf nfscrrjoocr fe uec sfrmf nf pjtfekmcs z₄ kjij kfetrj. Tfrj sjgrf uec smeluocrmncn cmsocnc z9z₄, fs pjsmgof rfprfsfetcr c h pjr uec sfrmf quf qu f mevjoukrc pjtfekmcs fetfrcs nf z-z₄ eflctmvcs y pjsmtmvcs, fs nfkmr0  

(3)

(3) sf pufnf fskrmgmr kjij oc suic nf njs sfrmfs0   y su pcrtf nf ocs pjtfekmcs eflctmvcs nf z-z₄, oocicnc pcrtf prmekmpco nf oc sfrmf (3) quf kjevfrlf fe |3/(z-z₄)|?r* j fqumvcofetf pcrc |3/(z-z₄)|5r*9r 0

 

Oc pcrtf kjesmstfetf ocs pjtfekmcs ej eflctmvcs e flctmvcs nf z--z₄ sf of oocic pcrtf cecomtmkc nf oc sfrmf (3) quf kjevfrlf pcrc |z--z₄|?W0 |z--z₄|?W0

Oc suic sjgrf ojs eflctmvjs y pjsmtmvjs nf (3) sf rffskrmgmrã fe uec hjric smipomkcnc kjij0

 

^fjrfic nf nf Ocurfet Ocurf et [fc h cecomtmkc fe fo njimemj nfo cemooj N quf sf nfef pjr r ? |z--z₄| ? W. Fetjekfs h sf rfprfsfetcréc fe sfrmf kjij0

vãomnc pcrc r ? |z--z₄| ? W. Ojs kjfkmfetfs cb ncnjs pjr0

njenf K fs uec kurvc ku rvc smipof kfrrcnc quf sf fekufetrc tjtcoifetf nfetrj nf N y quf tmfef c z₄ fe su metfrmjr, kjij sf iufstrc fe oc lurc

 

Nfijstrckmje nfo tfjrfic nf Ocurfet [fce K3 y K7 kmrkuehfrfekmcs kjekíetrmkcs kje kfetrjs fe z₄ y rcnmjs r3 y W7, njenf r ? r3 ? W7 ? W. [fc z ue puetj `j fe N, quf tcigmíe sctmshckf oc nfsmluconcn r3? |z-z₄| ? W7. [f metrjnukf ue kjrtf trcesvfrsco fe K7 y K3 (kjij sf iufstrc fe oc lurc), csé nfnukmfenj oc hjriuoc metflrco nf Kcukay0

 

Oc hjriuoc metflrco nf Kcukay sf pufnf rffskrmgmr nf oc smlumfetf icefrc0

 

Cajrc smfenj n9|z - z₄|, y I fo vcojr icxmij nf | h(z)| sjgrf fo kjetjrej K3. Vscenj |s - z₄|9 r3 y oc nfsmluconcn0 pjr oj tcetj IO fs0

pjr oj tcetj0

 

Fe rfsuife ◊ [fc z 9 z₄ uec smeluocrmncn cmsocnc nf oc huekmûe h (z). Ve nfscrrjooj fe [frmf

nf Ocurfet pcrc h (z) fs ue nfscrrjooj nf oc hjric0

njenf oc kjevfrlfekmc jkurrf fe fo cemooj r ? |z ∗z₄| ? W. Oc pcrtf fe rj`j sf oocic oc pcrtf prmekmpco nf oc sfrmf quf kjevfrlf pcrc |z ∗ zj| 5 r2 imfetrcs quf c oc pcrtf fe czuo sf of oocic oc pcrtf cecoétmkc nf oc sfrmf quf kjevfrlf pcrc |z ∗ zj| ? W.

 

◊ [fc h (z) uec huekmûe cecoétmkc nfetrj nfo njimemj ceuocr N nfemnj kjij

r ? |z ∗ z₄| ? W. Fetjekfs, h (z) tmfef uec rfprfsfetckm¾je fe sfrmf nf pjtfekmcs nf oc hjric0

vãomnc pcrc su njimemj ceuocr y njenf ojs kjfkmfetfs pufnfe sfr kcokuocnjs pjr oc hûriuoc

njenf K fs uec kurvc kur vc kfrrcnc smipof ojkcomzcnc nfetrj nf N y quf kjetmfef c z₄.

 

^jicr fe kufetc oj smlumfetf0 • Ojs kjfkmfetfs cb fstãe rfockmjecnjs kje ojs trfs nctjs nfo njimemj ceuocr0 fo kfetrj zj, fo rcnmj metfrej r y fo rcnmj fxtfrej W. • Ojs kjfkmfetfs cb fstãe ðemkcifetf nftfrimecnjs pjr fo njimemj ceuocr y pjr oc huekmûe h (z). • Oc hûriuoc nf oc metflrco pcrc nftfrimecr nftfrimecr ojs kjfkmfetfs cb rcrc vfz fs utmomzcnc fe oc prãktmkc2 tcofs nfscrrjoojs sje fe lfefrco nmhékmofs nf kcokuocr y sf rfkurrf c tíkemkcs colfgrc colfgrcmkcs. mkcs.