The purpose of this entry is to collect a table of Mittag-Leffler type partial fraction expansions for various functions.

Elementary Functions

\begin{eqnarray} \pi \cot (\pi z) &=& {1 \over z} + \sum_{n=1}^\infty \left( {1 \over z - n} + {1 \over z + n} \right) \\ \pi \sec (\pi z) &=& {2 \over 1 - 2z} + \sum_{k=1}^\infty (-1)^{k+1} \left( {2 \over 2k - 1 - 2z} - {2 \over 2k + 1 - 2z} \right) \\ \end{eqnarray}

Hypergeometric Functions

Gamma Functions

\begin{eqnarray} \psi (z) = {\Gamma'(z) \over \Gamma(z)} + \gamma &=& {1 \over z} + \sum_{k=1}^\infty \left( {1 \over k} - {1 \over z + k} \right) \\ (-1)^n {\psi^{(n)} (z) \over n!} &=& \sum_{k=0}^\infty {1 \over (z + k)^n} \\ {\Gamma (x) \Gamma(\frac{1}{2}) \over \Gamma (x + \frac{1}{2})} &=& \sum_{n=0}^\infty {(2n)! \over 2^{2n} (n!)^2} {1 \over x + n} \\ \end{eqnarray} Here $\gamma$ is Mascheroni's constant.

Elliptic Functions

\begin{eqnarray} \wp \left(z \left| \frac{1}{2} \omega, \frac{1}{2} \omega' \right.\right) &=& {1 \over z^2} + \sum_{|k| + |k'| \neq 0} \left( {1 \over (z - k \omega - k' \omega')^2} - {1 \over (k \omega + k' \omega')^2} \right) \\ \end{eqnarray}