| Version 5 |
Version 4 |
| The purpose of this entry is to collect a table of Mittag-Leffler |
The purpose of this entry is to collect a table of Mittag-Leffler |
| partial fraction expansions for various functions. |
partial fraction expansions for various functions. |
|
|
| \section{Elementary Functions} |
\section{Elementary Functions} |
|
|
| \begin{eqnarray} |
\begin{eqnarray} |
| \cot (\pi z) &=& {1 \over z} + \sum_{n=1}^\infty \left( {1 \over z - |
\cot (\pi z) &=& {1 \over z} + \sum_{n=1}^\infty \left( {1 \over z - |
| n} + {1 \over z + n} \right) \\ |
n} + {1 \over z + n} \right) \\ |
| \pi \sec (\pi z) &=& {2 \over 1 - 2z} + \sum_{k=1}^\infty (-1)^{k+1} |
\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) \\ |
\left( {2 \over 2k - 1 - 2z} - {2 \over 2k + 1 - 2z} \right) \\ |
| \end{eqnarray} |
\end{eqnarray} |
|
|
| \section{Hypergeometric Functions} |
\section{Hypergeometric Functions} |
| \begin{eqnarray} |
\begin{eqnarray} |
|
{}_2F_1 (z,1;z+1;w) &=& \sum_{k=0}^\infty {w^k \over z+k} \\
|
{}_2F_1 (z,1;z+1;w) &=& \sum_{k=0}^\infty {w^k \over z+k} |
| \end{eqnarray} |
\end{eqnarray} |
|
|
| \section{Elliptic Functions} |
\section{Elliptic Functions} |
|
|
| \begin{eqnarray} |
\begin{eqnarray} |
| \wp (z | \frac{1}{2} \omega, \frac{1}{2} \omega') &=& |
\wp (z | \frac{1}{2} \omega, \frac{1}{2} \omega') &=& |
| \sum_{|k| + |k'| \neq 0} \left( |
\sum_{|k| + |k'| \neq 0} \left( |
| {1 \over (z - k \omega - k' \omega')^2} - |
{1 \over (z - k \omega - k' \omega')^2} - |
| {1 \over (k \omega + k' \omega')^2} \right) \\ |
{1 \over (k \omega + k' \omega')^2} \right) \\ |
| \end{eqnarray} |
|
