PlanetMath: partial fraction series for digamma function

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partial fraction series for digamma function

(Theorem)

Theorem 1

$\displaystyle \psi (z) = - \gamma -\frac{1}{z} + \sum_{k=1}^\infty \left( \frac... ... k} \right)=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{z+k}\right)$

Proof: Start with

$\displaystyle \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^{z/k},$

$\displaystyle \ln\Gamma(z)=-\gamma z - \ln z +\sum_{k=1}^{\infty}\left(-\ln\left(1+\frac{z}{k}\right)+\frac{z}{k}\right)$

and thus, taking derivatives,

$\displaystyle \psi(z)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(-\frac{1/k}{... ... =-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{z+k}\right)$

The second formula follows after rearranging terms (the rearrangement is legal since we are simply exchanging adjacent terms, so partial sums remain the same).

"partial fraction series for digamma function" is owned by rm50.

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Cross-references: partial sums, adjacent, derivatives

This is version 3 of partial fraction series for digamma function, born on 2006-11-11, modified 2007-03-13.
Object id is 8540, canonical name is PartialFractionSeriesForDigammaFunction.
Accessed 672 times total.

Classification:

AMS MSC:	33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)
	30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

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