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