U matematici K funkciya zazvichaj poznachayetsya yak K z displaystyle K z uzagalnennya funkciyi giperfaktoriala dlya kompleksnih chisel podibno do gamma funkciyi yak uzagalnennya funkciyi faktoriala dlya kompleksnih chisel Oznachennya red Formalno K funkciya viznachayetsya tak K z 2p z 12exp Cz2 0z 1ln G t 1 dt displaystyle K z 2 pi frac z 1 2 exp left C z 2 int 0 z 1 ln Gamma t 1 dt right nbsp Takozh mozhna zapisati yiyi u prostishij formi K z exp z 1 z z 1 displaystyle K z exp bigl zeta 1 z zeta 1 bigr nbsp de z z displaystyle zeta z nbsp pohidni dzeta funkciyi Rimana z a z displaystyle zeta a z nbsp dzeta funkciya Gurvica i z a z def z s z s s a displaystyle zeta a z stackrel mathrm def left frac partial zeta s z partial s right s a nbsp Insha forma zapisu cherez poligamma funkciyu 1 K z exp ps 2 z z2 z2 z2ln 2p displaystyle K z exp left psi 2 z frac z 2 z 2 frac z 2 ln 2 pi right nbsp Abo vikoristovuyuchi uzagalnenu poligamma funkciyu en 2 mozhna skazati sho K z Aexp ps 2 z z2 z2 displaystyle K z A exp left psi 2 z frac z 2 z 2 right nbsp de A displaystyle A nbsp stala Glejshera Vlastivosti red Nehaj a gt 0 displaystyle alpha gt 0 nbsp Todi aa 1ln K x dx 01ln K x dx 12a2 ln a 12 displaystyle int alpha alpha 1 ln K x dx int 0 1 ln K x dx frac 1 2 alpha 2 left ln alpha frac 1 2 right nbsp Nehaj f a aa 1ln K x dx displaystyle f alpha int alpha alpha 1 ln K x dx nbsp Diferenciyuyuchi cyu rivnist po a displaystyle alpha nbsp otrimayemo f a ln K a 1 ln K a ln K a 1 K a displaystyle f alpha ln K alpha 1 ln K alpha ln frac K alpha 1 K alpha nbsp Za oznachennyam K funkciyi mozhna zapisati f a aln a displaystyle f alpha alpha ln alpha nbsp Takozh f a 12a2 ln a 12 C displaystyle f alpha frac 1 2 alpha 2 left ln alpha frac 1 2 right C nbsp Poklademo a 0 displaystyle alpha 0 nbsp Todi otrimayemo 01ln K x dx limt 0 12t2 ln t 12 C C displaystyle int 0 1 ln K x dx lim t to 0 left frac 1 2 t 2 left ln t frac 1 2 right right C C nbsp Teper mozhna zrobiti visnovok pro rivnist navedenu vishe K funkciya tisno pov yazana z gamma funkciyeyu ta G funkciyeyu Barnsa en dlya naturalnih n displaystyle n nbsp mayemo K n G n n 1G n displaystyle K n frac bigl Gamma n bigr n 1 G n nbsp Mozhna zapisati cyu rivnist bilsh prosto K n 1 11 22 33 nn displaystyle K n 1 1 1 cdot 2 2 cdot 3 3 cdots n n nbsp Znachennya funkciyi pri naturalnih argumentah 1 4 108 27648 86400000 4031078400000 3319766398771200000 displaystyle 1 4 108 27648 86400000 4031078400000 3319766398771200000 ldots quad nbsp poslidovnist A002109 z Onlajn enciklopediyi poslidovnostej cilih chisel OEIS Literatura red Victor S Adamchik PolyGamma Functions of Negative Order Olivier EspinosaVictor Hugo Moll A Generalized polygamma function Integral Transforms and Special Functions Vol 15 No 2 April 2004 pp 101 115 Otrimano z https uk wikipedia org w index php title K funkciya amp oldid 36450940
