Triga mma fu nkciya v matematici ye drugoyu z poligamma funkcij Yiyi poznachayut ps 1 z displaystyle psi 1 z i viznachayut yakTrigamma funkciya dijsnogo argumentu x ps 1 z d 2 d z 2 ln G z displaystyle psi 1 z frac rm d 2 rm d z 2 ln Gamma z de G z displaystyle Gamma z gamma funkciya 1 Z cogo viznachennya viplivaye sho ps 1 z d d z ps z displaystyle psi 1 z frac rm d rm d z psi z de ps z displaystyle psi z digamma funkciya persha z poligamma funkcij 2 Trigamma funkciyu mozhna takozh viznachiti cherez sumu takogo ryadu ps 1 z n 0 1 z n 2 displaystyle psi 1 z sum n 0 infty frac 1 z n 2 zvidki vidno sho vona ye okremim vipadkom dzeta funkciyi Gurvica ps 1 z z 2 z displaystyle psi 1 z zeta 2 z Ci formuli istinni koli z 0 1 2 3 displaystyle z neq 0 1 2 3 ldots u zaznachenih tochkah funkciya ps 1 z displaystyle psi 1 z maye kvadratichni singulyarnosti div grafik funkciyi Isnuyut takozh inshi poznachennya dlya ps 1 z displaystyle psi 1 z vikoristovuvani v literaturi ps z ps 1 z displaystyle psi z psi 1 z Inodi termin trigamma funkciya zastosovuyut dlya funkciyi F z ps 1 z 1 displaystyle displaystyle F z psi 1 z 1 1 Zmist 1 Integralni podannya 2 Inshi formuli 3 Chastkovi znachennya 4 Div takozh 5 Primitki 6 PosilannyaIntegralni podannya RedaguvatiVikoristovuyuchi podannya u viglyadi ryadu a takozh formulu dlya sumi chleniv geometrichnoyi progresiyi mozhna otrimati take podvijne integralne podannya ps 1 z 0 1 0 y x z 1 y 1 x d x d y displaystyle psi 1 z int 0 1 int 0 y frac x z 1 y 1 x rm d x rm d y nbsp Za dopomogoyu integruvannya za chastinami vihodit take odinarne podannya ps 1 z 0 1 x z 1 ln x 1 x d x displaystyle psi 1 z int 0 1 frac x z 1 ln x 1 x rm d x nbsp Vikoristovuyetsya takozh inshe podannya yake mozhna otrimati z poperednogo zaminoyu x e t ps 1 z 0 e z t 1 e t d t displaystyle psi 1 z int 0 infty frac e zt 1 e t rm d t nbsp Inshi formuli RedaguvatiTrigamma funkciya zadovolnyaye rekurentne spivvidnoshennya 2 ps 1 z 1 ps 1 z 1 z 2 displaystyle psi 1 z 1 psi 1 z frac 1 z 2 nbsp a takozh formulu dopovnennya ps 1 1 z ps 1 z p 2 sin 2 p z displaystyle psi 1 1 z psi 1 z frac pi 2 sin 2 pi z nbsp Dlya trigamma funkciyi kratnogo argumentu isnuye taka vlastivist 2 ps 1 k z 1 k 2 n 0 k 1 ps 1 z n k displaystyle psi 1 kz frac 1 k 2 sum n 0 k 1 psi 1 left z frac n k right nbsp Navedemo takozh asimptotichnij rozklad iz vikoristannyam chisel Bernulli ps 1 z 1 1 z 1 2 z 2 k 1 B 2 k z 2 k 1 displaystyle psi 1 z 1 frac 1 z frac 1 2z 2 sum k 1 infty frac B 2k z 2k 1 nbsp Chastkovi znachennya RedaguvatiNizhche navedeno chastkovi znachennya trigamma funkciyi ps 1 1 4 p 2 8 G displaystyle psi 1 left tfrac 1 4 right pi 2 8G nbsp ps 1 1 3 2 3 p 2 3 3 C l 2 2 3 p displaystyle psi 1 left tfrac 1 3 right tfrac 2 3 pi 2 3 sqrt 3 mathrm Cl 2 left tfrac 2 3 pi right nbsp ps 1 1 2 1 2 p 2 displaystyle psi 1 left tfrac 1 2 right tfrac 1 2 pi 2 nbsp ps 1 2 3 2 3 p 2 3 3 C l 2 2 3 p displaystyle psi 1 left tfrac 2 3 right tfrac 2 3 pi 2 3 sqrt 3 mathrm Cl 2 left tfrac 2 3 pi right nbsp ps 1 3 4 p 2 8 G displaystyle psi 1 left tfrac 3 4 right pi 2 8G nbsp ps 1 1 1 6 p 2 displaystyle psi 1 1 tfrac 1 6 pi 2 nbsp de G stala Katalana a C l 2 8 displaystyle mathrm Cl 2 theta nbsp funkciya Klauzena pov yazana z uyavnoyu chastinoyu dilogarifma cherez C l 2 8 I m L i 2 e i 8 displaystyle mathrm Cl 2 theta mathrm Im left mathrm Li 2 left e mathrm i theta right right nbsp Vikoristovuyuchi formulu kratnogo argumentu i formulu dopovnennya a takozh zv yazok ps 1 1 8 displaystyle psi 1 left tfrac 1 8 right nbsp z funkciyeyu Klauzena 3 4 mayemo ps 1 1 6 2 p 2 15 3 C l 2 2 3 p displaystyle psi 1 left tfrac 1 6 right 2 pi 2 15 sqrt 3 mathrm Cl 2 left tfrac 2 3 pi right nbsp ps 1 5 6 2 p 2 15 3 C l 2 2 3 p displaystyle psi 1 left tfrac 5 6 right 2 pi 2 15 sqrt 3 mathrm Cl 2 left tfrac 2 3 pi right nbsp ps 1 1 8 2 2 p 2 4 4 2 G 16 2 C l 2 p 4 displaystyle psi 1 left tfrac 1 8 right 2 sqrt 2 pi 2 4 4 sqrt 2 G 16 sqrt 2 mathrm Cl 2 left tfrac pi 4 right nbsp ps 1 3 8 2 2 p 2 4 4 2 G 16 2 C l 2 p 4 displaystyle psi 1 left tfrac 3 8 right 2 sqrt 2 pi 2 4 4 sqrt 2 G 16 sqrt 2 mathrm Cl 2 left tfrac pi 4 right nbsp ps 1 5 8 2 2 p 2 4 4 2 G 16 2 C l 2 p 4 displaystyle psi 1 left tfrac 5 8 right 2 sqrt 2 pi 2 4 4 sqrt 2 G 16 sqrt 2 mathrm Cl 2 left tfrac pi 4 right nbsp ps 1 7 8 2 2 p 2 4 4 2 G 16 2 C l 2 p 4 displaystyle psi 1 left tfrac 7 8 right 2 sqrt 2 pi 2 4 4 sqrt 2 G 16 sqrt 2 mathrm Cl 2 left tfrac pi 4 right nbsp Dlya znachen za mezhami intervalu 0 lt z 1 displaystyle 0 lt z leq 1 nbsp mozhna vikoristati rekurentne spivvidnoshennya navedene vishe Napriklad ps 1 5 4 p 2 8 G 16 displaystyle psi 1 left tfrac 5 4 right pi 2 8G 16 nbsp ps 1 3 2 1 2 p 2 4 displaystyle psi 1 left tfrac 3 2 right tfrac 1 2 pi 2 4 nbsp ps 1 2 1 6 p 2 1 displaystyle psi 1 2 tfrac 1 6 pi 2 1 nbsp Div takozh RedaguvatiGamma funkciya Digamma funkciya Poligamma funkciya Stala Katalana Dzeta funkciya GurvicaPrimitki Redaguvati a b Eric W Weisstein Trigamma funkciya angl na sajti Wolfram MathWorld a b v Eric W Weisstein Poligamma funkciya angl na sajti Wolfram MathWorld C C Grosjean Formulae concerning the computation of the Clausen integral C l 2 8 displaystyle mathrm Cl 2 theta nbsp J Comp Appl Math 11 1984 331 342 P J de Doelder On the Clausen integral C l 2 8 displaystyle mathrm Cl 2 theta nbsp and a related integral J Comp Appl Math 11 1984 325 330Posilannya RedaguvatiMilton Abramowitz amp Irene A Stegun Handbook of Mathematical Functions en 1964 Dover Publications New York ISBN 0 486 61272 4 Div rozdil 6 4 Arhivovano 2 veresnya 2009 u Wayback Machine Eric W Weisstein Trigamma funkciya angl na sajti Wolfram MathWorld Eric W Weisstein Poligamma funkciya angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Trigamma funkciya amp oldid 35095219
