Definition:Digamma Function

Definition

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.

Also known as

The digamma function is also known as the psi function.

Examples

Example: $\map \psi {\dfrac 1 6}$

$\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$

Example: $\map \psi {\dfrac 1 4}$

$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$

Example: $\map \psi {\dfrac 1 3}$

$\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

Example: $\map \psi {\dfrac 1 2}$

$\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$

Example: $\map \psi {\dfrac 2 3}$

$\map \psi {\dfrac 2 3} = -\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$

Example: $\map \psi {\dfrac 3 4}$

$\map \psi {\dfrac 3 4} = -\gamma - 3 \ln 2 + \dfrac \pi 2$

Example: $\map \psi {\dfrac 5 6}$

$\map \psi {\dfrac 5 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 + \dfrac {\pi \sqrt 3} 2$

Example: $\map \psi 1$

$\map \psi 1 = -\gamma$

Example: $\map \psi {\dfrac 3 2}$

$\map \psi {\dfrac 3 2} = -\gamma - 2 \ln 2 + 2$

Example: $\map \psi {\dfrac 4 3}$

$\map \psi {\dfrac 4 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3} + 3$

Example: $\map \psi {\dfrac 5 4}$

$\map \psi {\dfrac 5 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2 + 4$

Example: $\map \psi {\dfrac 7 6}$

$\map \psi {\dfrac 7 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2 + 6$

Example: $\map \psi 2$

$\map \psi 2 = -\gamma + 1$

Also see

• Results about the digamma function can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): digamma function or psi function
• Weisstein, Eric W. "Digamma Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DigammaFunction.html