PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] 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

\begin{eqnarray} \pi \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}

Hypergeometric Functions

\begin{eqnarray} {}_2F_1 (z,1;z+1;w) &=& \sum_{k=0}^\infty {w^k \over z+k} \\ \end{eqnarray}

Gamma Functions

\begin{eqnarray} \psi (z) = {\Gamma'(z) \over \Gamma(z)} + \gamma &=& {1 \over z} + \sum_{k=1}^\infty \left( {1 \over k} - {1 \over z + k} \right) \\ (-1)^n {\psi^{(n)} (z) \over n!} &=& \sum_{k=0}^\infty {1 \over (z + k)^n} \\ {\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.

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}



Anyone with an account can edit this entry. Please help improve it!

"table of partial fraction expansions" is owned by rspuzio. [ full author list (4) ]
(view preamble | get metadata)

View style:

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


This object's parent.
Log in to rate this entry.
(view current ratings)

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 2911 times total.

Classification:
AMS MSC30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)