PlanetMath: digamma and polygamma function

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (51)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

digamma and polygamma function

(Definition)

The digamma function is defined as the logarithmic derivative of the gamma function: $$ \psi (z) = {d \over dz} \log \Gamma (z) = {\Gamma' (z) \over \Gamma (z)}. $$ Likewise the polygamma functions are defined as higher order logarithmic derivatives of the gamma function: $$ \psi^{(n)} (z) = {d^n \over dz^n} \log \Gamma (z). $$

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

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

These functions have poles at the negative integers and can be expressed as partial fraction series:

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

(1)

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

(2)

Here, $\gamma$ is Euler-Mascheroni constant. Substituting $z = 1$ to (1), one gets the value $$\Gamma\,'(1) \;=\; -\gamma.$$

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

(view preamble | get metadata)

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

Log in to rate this entry.
(view current ratings)

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 3804 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)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

No messages.

Interact