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

 $\displaystyle\pi\cot(\pi z)$ $\displaystyle=$ $\displaystyle{1\over z}+\sum_{n=1}^{\infty}\left({1\over z-n}+{1\over z+n}\right)$ (1) $\displaystyle\pi\sec(\pi z)$ $\displaystyle=$ $\displaystyle{2\over 1-2z}+\sum_{k=1}^{\infty}(-1)^{k+1}\left({2\over 2k-1-2z}% -{2\over 2k+1-2z}\right)$ (2)

2 Hypergeometric Functions

 $\displaystyle{}_{2}F_{1}(z,1;z+1;w)$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{\infty}{w^{k}\over z+k}$ (4)

3 Gamma Functions

 $\displaystyle\psi(z)={\Gamma^{\prime}(z)\over\Gamma(z)}+\gamma$ $\displaystyle=$ $\displaystyle{1\over z}+\sum_{k=1}^{\infty}\left({1\over k}-{1\over z+k}\right)$ (6) $\displaystyle(-1)^{n}{\psi^{(n)}(z)\over n!}$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{\infty}{1\over(z+k)^{n}}$ (7) $\displaystyle{\Gamma(x)\Gamma(\frac{1}{2})\over\Gamma(x+\frac{1}{2})}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}{(2n)!\over 2^{2n}(n!)^{2}}{1\over x+n}$ (8)

Here $\gamma$ is Mascheroni’s constant.

4 Elliptic Functions

 $\displaystyle\wp\left(z\left|\frac{1}{2}\omega,\frac{1}{2}\omega^{\prime}% \right.\right)$ $\displaystyle=$ $\displaystyle{1\over z^{2}}+\sum_{|k|+|k^{\prime}|\neq 0}\left({1\over(z-k% \omega-k^{\prime}\omega^{\prime})^{2}}-{1\over(k\omega+k^{\prime}\omega^{% \prime})^{2}}\right)$ (10)
Title table of partial fraction expansions TableOfPartialFractionExpansions 2013-03-22 15:44:29 2013-03-22 15:44:29 rspuzio (6075) rspuzio (6075) 20 rspuzio (6075) Example msc 30D30 ElementaryFunction GammaFunction HypergeometricFunction EllipticFunction