| Version 10 |
Version 9 |
| The purpose of this entry is to collect a table of Mittag-Leffler type |
The purpose of this entry is to collect a table of Mittag-Leffler type |
| 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{Gamma Functions} |
\section{Gamma Functions} |
|
|
| \begin{eqnarray} |
\begin{eqnarray} |
| {\Gamma'(z) \over Gamma(z)} &=& - \gamma + {1 \over z} + |
{\Gamma'(z) \over Gamma(z)} &=& - \gamma + {1 \over z} + |
| \sum_{k=1}^\infty \left( {1 \over n} - {1 \over z + k} \right) \\ |
\sum_{k=1}^\infty \left( {1 \over n} - {1 \over z + k} \right) \\ |
| {\Gamma (x) \Gamma(\frac{1}{2}) \over \Gamma (x + \frac{1}{2})} &=& |
{\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} \\
|
\sum_{n=0}^\infty {(2n)! \over 2^(2n) (n!)^2} {1 \over x + n} \\
|
| \end{eqnarray} |
\end{eqnarray} |
|
|
| Here $\gamma$ is Mascheroni's constant. |
Here $\gamma$ is Mascheroni's constant. |
|
|
| \section{Elliptic Functions} |
\section{Elliptic Functions} |
|
|
| \begin{eqnarray} |
\begin{eqnarray} |
| \wp \left(z \left| \frac{1}{2} \omega, \frac{1}{2} \omega' \right.\right) &=& |
\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^2} + \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} |
\end{eqnarray} |
