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

$\pi \cot (\pi z)$ $=$ ${\displaystyle \frac{1}{z}}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{z-n}}+{\displaystyle \frac{1}{z+n}}\right)$ (1)

$\pi \sec (\pi z)$ $=$ ${\displaystyle \frac{2}{1-2z}}+{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^{k+1}\left({\displaystyle \frac{2}{2k-1-2z}}-{\displaystyle \frac{2}{2k+1-2z}}\right)$ (2)

2 Hypergeometric Functions

${}_{2}F_1(z,1;z+1;w)$ $=$ ${\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{w^k}{z+k}}$ (4)

3 Gamma Functions

$\psi (z)={\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }(z)}{\mathrm{\Gamma }(z)}}+\gamma$ $=$ ${\displaystyle \frac{1}{z}}+{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{z+k}}\right)$ (6)

${(-1)}^n{\displaystyle \frac{{\psi }^{(n)}(z)}{n!}}$ $=$ ${\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{1}{{(z+k)}^n}}$ (7)

${\displaystyle \frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(\frac{1}{2})}{\mathrm{\Gamma }(x+\frac{1}{2})}}$ $=$ ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{(2n)!}{2^{2n}{(n!)}^2}}{\displaystyle \frac{1}{x+n}}$ (8)

Here $\gamma$ is Mascheroni’s constant.

4 Elliptic Functions

$\mathrm{\wp }\left(z\right|{\displaystyle \frac{1}{2}}\omega ,{\displaystyle \frac{1}{2}}{\omega }^{\prime })$ $=$ ${\displaystyle \frac{1}{z^2}}+{\displaystyle \sum _{|k|+|k^{\prime }|\ne 0}}\left({\displaystyle \frac{1}{{(z-k\omega -k^{\prime }{\omega }^{\prime })}^2}}-{\displaystyle \frac{1}{{(k\omega +k^{\prime }{\omega }^{\prime })}^2}}\right)$ (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