PlanetMath: digamma and polygamma function

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digamma and polygamma function

(Definition)

The digamma function is defined as the logarithmic derivative of the gamma function:

$\displaystyle \psi (z) = {d \over dz} \log \Gamma (z) = {\Gamma' (z) \over \Gamma (z)}.$

Likewise the polygamma functions are defined as higher order logarithmic derivatives of the gamma function:

$\displaystyle \psi^{(n)} (z) = {d^n \over dz^n} \log \Gamma (z).$

These equations enjoy functional equations which are closely related to those of the gamma function:

$\displaystyle \psi (z+1)$	$\displaystyle = \psi (z) + {1 \over z}$
$\displaystyle \psi (1-z)$	$\displaystyle = \psi (z) + \pi \cot \pi z$
$\displaystyle \psi (2z)$	$\displaystyle = {1 \over 2} \psi (z) + {1 \over 2} \psi \left( z + {1 \over 2} \right) + \log 2$
$\displaystyle \psi^{(n)} (z+1)$	$\displaystyle = \psi^{(n)} (z) + (-1)^n {n! \over z^{n+1}}$

$\displaystyle \psi(z) = -\gamma-{1\over z}+\sum_{k=1}^\infty \left({1\over k}-{1\over z\!+\!k}\right),$

(1)

$\displaystyle \psi^{(n)}(z) = (-1)^nn!\!\sum_{k=0}^\infty{1\over(z\!+\!k)^n}$

(2)

Here, $\gamma$ is Euler-Mascheroni constant. Substituting

to (1), one gets the value

$\displaystyle \Gamma\,'(1) \;=\; -\gamma.$

"digamma and polygamma function" is owned by rspuzio. [ full author list (2) ]

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Also defines:

digamma function, polygamma function

This object's parent.

Attachments:: partial fraction series for digamma function (Theorem) by rm50
Gauss' digamma theorem (Theorem) by rm50

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Cross-references: series, partial fraction, integers, negative, poles, functions, functional equations, equations, order, gamma function, logarithmic derivative
There are 3 references to this entry.

This is version 10 of digamma and polygamma function, born on 2006-05-01, modified 2008-09-25.
Object id is 7890, canonical name is DigammaAndPolygammaFunction.
Accessed 2828 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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