# digamma and polygamma function

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

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

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)
 $\displaystyle\psi^{(n)}(z)=(-1)^{n}n!\!\sum_{k=0}^{\infty}{1\over(z\!+\!k)^{n}}$ (2)

Here, $\gamma$ is Euler–Mascheroni constant (http://planetmath.org/EulerMascheroniConstant).  Substituting  $z=1$  to (1), one gets the value

 $\Gamma\,^{\prime}(1)\;=\;-\gamma.$
Title digamma and polygamma function DigammaAndPolygammaFunction 2013-03-22 15:53:21 2013-03-22 15:53:21 rspuzio (6075) rspuzio (6075) 13 rspuzio (6075) Definition msc 30D30 msc 33B15 digamma function polygamma function