Hi funkciya Lezhandra ce specialna funkciya nazvana im yam francuzkogo matematika Adriyen Mari Lezhandra Viznachayetsya ryadom Tejlora yakij takozh ye ryadom Dirihle x n z k 0 z 2 k 1 2 k 1 n displaystyle chi nu z sum k 0 infty frac z 2k 1 2k 1 nu Takim chinom Hi funkciya Lezhandra trivialno virazhayetsya cherez polilogarifm x n z 1 2 Li n z Li n z displaystyle chi nu z frac 1 2 left operatorname Li nu z operatorname Li nu z right Hi funkciya Lezhandra vinikaye v diskretnomu peretvorenni Fur ye za indeksom n dzeta funkciyi Gurvica a takozh mnogochleniv Ejlera Hi funkciya Lezhandra ye okremim vipadkom dzeta funkciyi Lerha en x n z 2 n z F z 2 n 1 2 displaystyle chi n z 2 n z Phi z 2 n 1 2 Zmist 1 Totozhnosti 2 Integralni spivvidnoshennya 3 Literatura 4 PosilannyaTotozhnosti red x 2 x x 2 1 x p 2 4 i p 2 ln x displaystyle chi 2 x chi 2 1 x frac pi 2 4 frac i pi 2 ln x nbsp d d x x 2 x a r c t a n h x x displaystyle frac d dx chi 2 x frac rm arctanh x x nbsp Integralni spivvidnoshennya red 0 p 2 arcsin r sin 8 d 8 x 2 r displaystyle int 0 pi 2 arcsin r sin theta d theta chi 2 left r right nbsp 0 p 2 arctan r sin 8 d 8 1 2 0 p r 8 cos 8 1 r 2 sin 2 8 d 8 2 x 2 1 r 2 1 r displaystyle int 0 pi 2 arctan r sin theta d theta frac 1 2 int 0 pi frac r theta cos theta 1 r 2 sin 2 theta d theta 2 chi 2 left frac sqrt 1 r 2 1 r right nbsp 0 p 2 arctan p sin 8 arctan q sin 8 d 8 p x 2 1 p 2 1 p 1 q 2 1 q displaystyle int 0 pi 2 arctan p sin theta arctan q sin theta d theta pi chi 2 left frac sqrt 1 p 2 1 p cdot frac sqrt 1 q 2 1 q right nbsp 0 a 0 b d x d y 1 x 2 y 2 x 2 a b yaksho a b 1 displaystyle int 0 alpha int 0 beta frac dxdy 1 x 2 y 2 chi 2 alpha beta text yaksho alpha beta leq 1 nbsp Literatura red Djurdje Cvijovic Jacek Klinowski Values of the Legendre chi and Hurwitz zeta functions at rational arguments Math Comp 1999 No 68 P 1623 1630 Djurdje Cvijovic Integral representations of the Legendre chi function Journal of Mathematical Analysis and Applications 2006 Vol 2 no 332 P 1056 1062 ISSN 0022247X Posilannya red Weisstein Eric W Hi funkciya Lezhandra angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Hi funkciya Lezhandra amp oldid 38423573
