Eliptichni funkciyi Veyershtrassa odni z najprostishih eliptichnih funkcij Cej klas funkcij nazvanij na chest Karla Veyershtrassa Takozh yih nazivayut displaystyle wp funkciyami Veyershtrassa i vikoristovuyut dlya yih poznachennya simvol displaystyle wp stilizovane P Zmist 1 Viznachennya 1 1 Varianti viznachennya 2 Vlastivosti 3 Diferencialni i integralni rivnyannya 3 1 Diferencialni rivnyannya 3 2 Integralni rivnyannya 4 Dodatkovi vlastivosti 5 Virazhennya dovilnih eliptichnih funkcij cherez funkciyi Veyershtrassa 6 Div takozh 7 LiteraturaViznachennya RedaguvatiNehaj zadana deyaka gratka G displaystyle Gamma nbsp v C displaystyle mathbb C nbsp Todi displaystyle wp nbsp funkciyeyu Veyershtrassa na nij nazivayetsya meromorfna funkciya zadana yak suma ryadu E z 1 z 2 w G 0 1 z w 2 1 w 2 displaystyle wp E z frac 1 z 2 sum w in Gamma setminus 0 left frac 1 z w 2 frac 1 w 2 right nbsp Mozhna pobachiti sho taka funkciya bude G displaystyle Gamma nbsp periodichnoyu na C displaystyle mathbb C nbsp i tomu ye meromorfnoyu funkciyeyu na E displaystyle E nbsp Ryad sho zadaye funkciyu Veyershtrassa ye v pevnomu znachenni regulyarizovanoyu versiyeyu rozbizhnogo ryadu w G 1 z w 2 displaystyle sum w in Gamma frac 1 z w 2 nbsp nayivnoyi sprobi zadati G displaystyle Gamma nbsp periodichnu funkciyu Cej ryad ye absolyutno rozbizhnim a za vidsutnosti prirodnogo poryadku na G displaystyle Gamma nbsp maye sens govoriti tilki pro absolyutnu zbizhnist pri vsih z oskilki pri fiksovanomu z i pri velikih w moduli jogo chleniv povodyatsya yak 1 w 2 displaystyle frac 1 w 2 nbsp a suma w G 1 w 2 displaystyle sum w in Gamma frac 1 w 2 nbsp po dvovimirnih gratkah G displaystyle Gamma nbsp ye rozbizhnoyu Varianti viznachennya Redaguvati Zadayuchi gratku G displaystyle Gamma nbsp yiyi bazisom G m w 1 n w 2 m n Z displaystyle Gamma m omega 1 n omega 2 mid m n in mathbb Z nbsp mozhna zapisati z w 1 w 2 1 z 2 m n Z 2 0 0 1 z m w 1 n w 2 2 1 m w 1 n w 2 2 displaystyle wp z omega 1 omega 2 frac 1 z 2 sum m n in mathbb Z 2 setminus 0 0 left frac 1 z m omega 1 n omega 2 2 frac 1 m omega 1 n omega 2 2 right nbsp Takozh oskilki funkciya Veyershtrassa yak funkciya troh zminnih odnoridna a z a w 1 a w 2 a 2 w p z w 1 w 2 displaystyle wp az a omega 1 a omega 2 a 2 wp z omega 1 omega 2 nbsp poznachivshi t w 2 w 1 displaystyle tau omega 2 omega 1 nbsp maye misce rivnist z w 1 w 2 w 1 2 z w 1 1 t displaystyle wp z omega 1 omega 2 omega 1 2 wp z omega 1 1 tau nbsp Tomu rozglyadayut z t z 1 t 1 z 2 m n Z 2 0 0 1 z m n t 2 1 m n t 2 displaystyle wp z tau wp z 1 tau frac 1 z 2 sum m n in mathbb Z 2 setminus 0 0 left frac 1 z m n tau 2 frac 1 m n tau 2 right nbsp Vlastivosti RedaguvatiFunkciya Veyershtrassa E E C displaystyle wp E E mapsto widehat mathbb C nbsp parna meromorfna funkciya z yedinim polyusom drugogo poryadku v tochci 0 Skoristavshis rozkladom 1 w z 2 1 w 2 j 1 j 1 w j 2 z j displaystyle frac 1 w z 2 frac 1 w 2 sum nolimits j 1 infty frac j 1 w j 2 z j nbsp i posumuvavshi po w G 0 displaystyle w in Gamma setminus 0 nbsp mozhna oderzhati rozklad v tochci z 0 displaystyle z 0 nbsp funkciyi Veyershtrassa v ryad Lorana E z 1 z 2 k 2 2 k 1 G 2 k G z 2 k 2 displaystyle wp E z frac 1 z 2 sum k 2 infty 2k 1 G 2k Gamma z 2k 2 nbsp de G 2 k G w G 0 w 2 k displaystyle G 2k Gamma sum w in Gamma setminus 0 w 2k nbsp ryadi Ejzenshtejna dlya gratki G displaystyle Gamma nbsp vidpovidni neparni sumi rivni nulyu Prote koeficiyenti pri z 2 displaystyle z 2 nbsp i z 4 displaystyle z 4 nbsp chasto zapisuyut v inshij tradicijnij formi E z 1 z 2 1 20 g 2 G z 2 1 28 g 3 G z 4 displaystyle wp E z frac 1 z 2 frac 1 20 g 2 Gamma z 2 frac 1 28 g 3 Gamma z 4 dots nbsp de g 2 displaystyle g 2 nbsp i g 3 displaystyle g 3 nbsp modulyarni invarianti gratki G displaystyle Gamma nbsp g 2 G 60 G 4 G g 3 G 140 G 6 G displaystyle g 2 Gamma 60G 4 Gamma quad g 3 Gamma 140G 6 Gamma nbsp Diferencialni i integralni rivnyannya RedaguvatiDiferencialni rivnyannya Redaguvati Z viznachenimi ranishe poznachennyami funkciya zadovolnyaye nastupne diferencialne rivnyannya z 2 4 z 3 g 2 z g 3 displaystyle wp z 2 4 wp z 3 g 2 wp z g 3 nbsp Integralni rivnyannya Redaguvati Eliptichni funkciyi Veyershtrassa mozhut buti podani cherez obertannya eliptichnih integraliv Nehaj u y d s 4 s 3 g 2 s g 3 displaystyle u int y infty frac ds sqrt 4s 3 g 2 s g 3 nbsp de g2 i g3 prijmayutsya konstantami Todi y u displaystyle y wp u nbsp Dodatkovi vlastivosti RedaguvatiDlya eliptichnih funkcij Veyershtrassa vikonuyetsya det z z 1 y y 1 z y z y 1 0 displaystyle det begin bmatrix wp z amp wp z amp 1 wp y amp wp y amp 1 wp z y amp wp z y amp 1 end bmatrix 0 nbsp abo v bilsh simetrichnij formi det u u 1 v v 1 w w 1 0 displaystyle det begin bmatrix wp u amp wp u amp 1 wp v amp wp v amp 1 wp w amp wp w amp 1 end bmatrix 0 nbsp de u v w 0 displaystyle u v w 0 nbsp Takozh z y 1 4 z y z y 2 z y displaystyle wp z y frac 1 4 left frac wp z wp y wp z wp y right 2 wp z wp y nbsp i 2 z 1 4 z z 2 2 z displaystyle wp 2z frac 1 4 left frac wp z wp z right 2 2 wp z nbsp yaksho 2 z displaystyle 2z nbsp ne ye periodom Virazhennya dovilnih eliptichnih funkcij cherez funkciyi Veyershtrassa RedaguvatiBud yaka eliptichna funkciya z periodami a displaystyle a nbsp i b displaystyle b nbsp mozhe buti predstavlena u viglyadi f z h z g z z displaystyle f z h wp z g wp z wp z nbsp de h g racionalni funkciyi z displaystyle wp z nbsp funkciya Veyershtrassa z timi zh periodami sho i u f z displaystyle f z nbsp Yaksho pri comu f z displaystyle f z nbsp ye parnoyu funkciyeyu to yiyi mozhna predstaviti u viglyadi f z h z displaystyle f z h wp z nbsp de h racionalna Inshimi slovami pole eliptichnih funkcij z fundamentalnimi periodami a displaystyle a nbsp i b displaystyle b nbsp ye skinchennim rozshirennyam polya C displaystyle mathbb C nbsp kompleksnih chisel z porodzhuyuchimi elementami z displaystyle wp z nbsp i z displaystyle wp z nbsp Div takozh RedaguvatiEliptichna funkciya Eliptichna kriva Eliptichni funkciyi Yakobi Ryadi EjzenshtejnaLiteratura RedaguvatiK Chandrasekharan Elliptic functions 1980 Springer Verlag ISBN 0 387 15295 4 Serge Lang Elliptic Functions 1973 Addison Wesley ISBN 0 201 04162 6 Otrimano z https uk wikipedia org w index php title Eliptichni funkciyi Veyershtrassa amp oldid 30465613
