AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Contacts

ExactPolygamma(PeterO.Numbers.EInteger,PeterO.Numbers.ERational)

Method (no overloads)

Summary

psi^(n)(q) exactly, for n = 0 or an odd order n and a whole or half-whole
q other than a pole, or null: a rational times pi^(n + 1), plus a
rational, or for the digamma function a rational less Euler's constant and, at a half-whole
q, 2 ln 2.

Remarks

psi(1) = -gamma and psi(1/2) = -gamma - 2 ln 2 (DLMF 5.4.12, 5.4.13), and
psi^(n)(1) = n! zeta(n + 1) and psi^(n)(1/2) = (2^(n + 1) - 1) n! zeta(n + 1) for an odd
n (DLMF 5.15.2, 5.15.3), where zeta(2m) = (-1)^(m + 1) B_(2m) (2 pi)^(2m)/(2 (2m)!) is a
rational times pi^(2m); the recurrence psi^(n)(z + 1) = psi^(n)(z) + (-1)^n n!/z^(n + 1) moves the argument to either. An even order from 2 wants zeta at an odd number, which has no
closed form here. https://github.com/asc-community/AngouriMath/issues/171

AngouriMath

Table of contents

AngouriMath

ExactPolygamma​(PeterO.​Numbers.​EInteger,​PeterO.​Numbers.​ERational)

ExactPolygamma(PeterO.Numbers.EInteger,PeterO.Numbers.ERational)