table of partial fraction expansionsTableOfMittagLefflerPartialFractionExpansions2006-03-06 22:40:452006-10-28 20:05:16ExampleMittag-Leffler's theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe purpose of this entry is to collect a table of Mittag-Leffler type
partial fraction expansions for various functions.
\section{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}
\section{Hypergeometric Functions}
\begin{eqnarray}
{}_2F_1 (z,1;z+1;w) &=& \sum_{k=0}^\infty {w^k \over z+k} \\
\end{eqnarray}
\section{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.
\section{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}