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table of partial fraction expansions

(Example)

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

Elementary Functions

$\displaystyle \pi \cot (\pi z)$	$\displaystyle =$	$\displaystyle {1 \over z} + \sum_{n=1}^\infty \left( {1 \over z - n} + {1 \over z + n} \right)$	(1)
$\displaystyle \pi \sec (\pi z)$	$\displaystyle =$	$\displaystyle {2 \over 1 - 2z} + \sum_{k=1}^\infty (-1)^{k+1} \left( {2 \over 2k - 1 - 2z} - {2 \over 2k + 1 - 2z} \right)$	(2)

Hypergeometric Functions

$\displaystyle {}_2F_1 (z,1;z+1;w)$

$\displaystyle =$

$\displaystyle \sum_{k=0}^\infty {w^k \over z+k}$

(3)

Gamma Functions

$\displaystyle \psi (z) = {\Gamma'(z) \over \Gamma(z)} + \gamma$	$\displaystyle =$	$\displaystyle {1 \over z} + \sum_{k=1}^\infty \left( {1 \over k} - {1 \over z + k} \right)$	(4)
$\displaystyle (-1)^n {\psi^{(n)} (z) \over n!}$	$\displaystyle =$	$\displaystyle \sum_{k=0}^\infty {1 \over (z + k)^n}$	(5)
$\displaystyle {\Gamma (x) \Gamma(\frac{1}{2}) \over \Gamma (x + \frac{1}{2})}$	$\displaystyle =$	$\displaystyle \sum_{n=0}^\infty {(2n)! \over 2^{2n} (n!)^2} {1 \over x + n}$	(6)

Here $\gamma$ is Mascheroni's constant.

Elliptic Functions

$\displaystyle \wp \left(z \left\vert \frac{1}{2} \omega, \frac{1}{2} \omega' \right.\right)$

$\displaystyle =$

$\displaystyle {1 \over z^2} + \sum_{\vert k\vert + \vert k'\vert \neq 0} \left(... ...er (z - k \omega - k' \omega')^2} - {1 \over (k \omega + k' \omega')^2} \right)$

(7)

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"table of partial fraction expansions" is owned by rspuzio. [ full author list (4) ]

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See Also: elementary function, gamma function, hypergeometric function, elliptic function

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Cross-references: Mascheroni's constant, functions, partial fraction, type

This is version 17 of table of partial fraction expansions, born on 2006-03-06, modified 2006-10-28.
Object id is 7692, canonical name is TableOfMittagLefflerPartialFractionExpansions.
Accessed 1920 times total.

Classification:

AMS MSC:

30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

Pending Errata and Addenda

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