Properties of Digamma Function

Theorem

This page gathers together some of the properties of the digamma function:

The digamma function, $\psi$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the logarithmic derivative of the gamma function:

$\map \psi z = \dfrac {\map {\Gamma'} z} {\map \Gamma z}$

where $\Gamma$ is the gamma function, and $\Gamma'$ denotes its derivative.

Digamma Function in terms of General Harmonic Number

$\ds \map \psi {z + 1} = -\gamma + \harm 1 z$

Recurrence Relation for Digamma Function

$\map \psi {z + 1} = \map \psi z + \dfrac 1 z$

Digamma Reflection Formula

$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$

Digamma Additive Formula

$\ds \map \psi {n z} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n$

Gauss's Digamma Theorem

$\ds \map \psi {\dfrac p q} = -\gamma - \ln 2 q - \dfrac \pi 2 \map \cot {\dfrac p q \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {q / 2} - 1} \map \cos {\dfrac {2 \pi p n} q} \map \ln {\map \sin {\dfrac {\pi n} q} }$

Ramanujan Phi Function in terms of Digamma Function

$\map \psi {\dfrac 1 z} + \map \psi {1 - \dfrac 1 z} = -2 \gamma - z \map \phi z$

Gauss's Integral Form of Digamma Function

$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$

Dirichlet's Integral Form of Digamma Function

$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z } } \rd t$