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'table of partial fraction expansions'
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| Title of object: |
table of partial fraction expansions |
| Canonical Name: |
TableOfMittagLefflerPartialFractionExpansions |
| Type: |
Example |
| Created on: |
2006-03-06 22:40:45 |
| Modified on: |
2006-03-25 13:30:03 |
| Classification: |
msc:30D30 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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% define commands here |
Content:
The purpose of this entry is to collect a table of Mittag-Leffler type
partial fraction expansions for various functions.
\section{Elementary Functions}
\begin{eqnarray}
\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}
{\Gamma'(z) \over \Gamma(z)} &=& - \Gamma + {1 \over z} +
\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})} &=&
\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} |
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