table of partial fraction expansions
The purpose of this entry is to collect a table of Mittag-Leffler type partial fraction expansions for various functions.
1 Elementary Functions
πcot(πz) | = | 1z+∞∑n=1(1z-n+1z+n) | (1) | ||
πsec(πz) | = | 21-2z+∞∑k=1(-1)k+1(22k-1-2z-22k+1-2z) | (2) |
2 Hypergeometric Functions
F12(z,1;z+1;w) | = | ∞∑k=0wkz+k | (4) |
3 Gamma Functions
ψ(z)=Γ′(z)Γ(z)+γ | = | 1z+∞∑k=1(1k-1z+k) | (6) | ||
(-1)nψ(n)(z)n! | = | ∞∑k=01(z+k)n | (7) | ||
Γ(x)Γ(12)Γ(x+12) | = | ∞∑n=0(2n)!22n(n!)21x+n | (8) |
Here γ is Mascheroni’s constant.
4 Elliptic Functions
℘(z|12ω,12ω′) | = | 1z2+∑|k|+|k′|≠0(1(z-kω-k′ω′)2-1(kω+k′ω′)2) | (10) |
Title | table of partial fraction expansions |
---|---|
Canonical name | TableOfPartialFractionExpansions |
Date of creation | 2013-03-22 15:44:29 |
Last modified on | 2013-03-22 15:44:29 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 20 |
Author | rspuzio (6075) |
Entry type | Example |
Classification | msc 30D30 |
Related topic | ElementaryFunction |
Related topic | GammaFunction |
Related topic | HypergeometricFunction |
Related topic | EllipticFunction |