table of partial fraction expansions

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

1 Elementary Functions

$\pi \cot (\pi z)$$=$${\displaystyle \frac{1}{z}}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{z-n}}+{\displaystyle \frac{1}{z+n}}\right)$(1)

$\pi \sec (\pi z)$$=$${\displaystyle \frac{2}{1-2z}}+{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^{k+1}\left({\displaystyle \frac{2}{2k-1-2z}}-{\displaystyle \frac{2}{2k+1-2z}}\right)$(2)

2 Hypergeometric Functions

${}_{2}F_1(z,1;z+1;w)$$=$${\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{w^k}{z+k}}$(4)

3 Gamma Functions

$\psi (z)={\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }(z)}{\mathrm{\Gamma }(z)}}+\gamma$$=$${\displaystyle \frac{1}{z}}+{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{z+k}}\right)$(6)

${(-1)}^n{\displaystyle \frac{{\psi }^{(n)}(z)}{n!}}$$=$${\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{1}{{(z+k)}^n}}$(7)

${\displaystyle \frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(\frac{1}{2})}{\mathrm{\Gamma }(x+\frac{1}{2})}}$$=$${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{(2n)!}{2^{2n}{(n!)}^2}}{\displaystyle \frac{1}{x+n}}$(8)

Here $\gamma$ is Mascheroni’s constant.

4 Elliptic Functions

$\mathrm{\wp }\left(z\right|{\displaystyle \frac{1}{2}}\omega ,{\displaystyle \frac{1}{2}}{\omega }^{\prime })$$=$${\displaystyle \frac{1}{z^2}}+{\displaystyle \sum _{|k|+|k^{\prime }|\ne 0}}\left({\displaystyle \frac{1}{{(z-k\omega -k^{\prime }{\omega }^{\prime })}^2}}-{\displaystyle \frac{1}{{(k\omega +k^{\prime }{\omega }^{\prime })}^2}}\right)$(10)