# Specific values of the digamma function

Related topic: Digamma function

## Zeros

Table of $x_{n}$ to 50 digits for $0 \le n \le 10$
$\psi\!\left(x_{n}\right) = 0$

## Values at integers

$\psi\!\left(1\right) = -\gamma$
$\psi\!\left(2\right) = 1 - \gamma$
$\psi\!\left(3\right) = \frac{3}{2} - \gamma$
$\psi\!\left(n\right) = H_{n - 1} - \gamma$

## Values at simple fractions

$\psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma$
$\psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2}$
$\psi\!\left(\frac{2}{3}\right) = \frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2}$
$\psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2)$
$\psi\!\left(\frac{3}{4}\right) = \frac{\pi}{2} - \gamma - 3 \log(2)$
$\psi\!\left(\frac{1}{6}\right) = -\frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2}$
$\psi\!\left(\frac{5}{6}\right) = \frac{\sqrt{3} \pi}{2} - \gamma - 2 \log(2) - \frac{3 \log(3)}{2}$
$\psi\!\left(\frac{1}{8}\right) = -\frac{\pi}{2} \left(\sqrt{2} + 1\right) - \gamma - 4 \log(2) - \frac{\log\!\left(2 + \sqrt{2}\right) - \log\!\left(2 - \sqrt{2}\right)}{\sqrt{2}}$

## Values at general fractions

$\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)$

## Values of polygamma functions at integers and simple fractions

$\psi'\!\left(1\right) = \frac{{\pi}^{2}}{6}$
$\psi''\!\left(1\right) = -2 \zeta\!\left(3\right)$
$\psi^{(m)}\!\left(1\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1\right)$
$\psi'\!\left(2\right) = \frac{{\pi}^{2}}{6} - 1$
$\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)$
$\psi'\!\left(\frac{1}{2}\right) = \frac{{\pi}^{2}}{2}$
$\psi''\!\left(\frac{1}{2}\right) = -14 \zeta\!\left(3\right)$
$\psi'''\!\left(\frac{1}{2}\right) = {\pi}^{4}$
$\psi^{(m)}\!\left(\frac{1}{2}\right) = {\left(-1\right)}^{m + 1} \left({2}^{m + 1} - 1\right) m ! \zeta\!\left(m + 1\right)$
$\psi'\!\left(\frac{1}{4}\right) = {\pi}^{2} + 8 G$
$\psi'\!\left(\frac{3}{4}\right) = {\pi}^{2} - 8 G$
$\psi''\!\left(\frac{1}{4}\right) = -2 {\pi}^{3} - 56 \zeta\!\left(3\right)$
$\psi''\!\left(\frac{3}{4}\right) = 2 {\pi}^{3} - 56 \zeta\!\left(3\right)$
$\psi''\!\left(\frac{1}{6}\right) = -182 \zeta\!\left(3\right) - 4 \sqrt{3} {\pi}^{3}$
$\psi''\!\left(\frac{5}{6}\right) = -182 \zeta\!\left(3\right) + 4 \sqrt{3} {\pi}^{3}$

## Specific complex parts

$\operatorname{Im}\!\left(\psi\!\left(i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) + \frac{1}{2 y}$
$\operatorname{Im}\!\left(\psi\!\left(1 + i y\right)\right) = \frac{\pi}{2} \coth\!\left(\pi y\right) - \frac{1}{2 y}$
$\operatorname{Im}\!\left(\psi\!\left(\frac{1}{2} + i y\right)\right) = \frac{\pi}{2} \tanh\!\left(\pi y\right)$

## Limits at singularities

$\psi\!\left(-n\right) = {\tilde \infty}$
$\psi^{(m)}\!\left(-n\right) = {\tilde \infty}$
$\psi^{(m)}\!\left(\infty\right) = \lim_{x \to \infty} \psi\!\left(x\right) = \begin{cases} \infty, & m = 0\\0, & m > 0\\ \end{cases}$
$\lim_{x \to {0}^{+}} \psi\!\left(x\right) = {\left(-1\right)}^{m + 1} \infty$

## Infinite sums over zeros

$\sum_{n=0}^{\infty} \frac{1}{x_{n}^{2}} = {\gamma}^{2} + \frac{{\pi}^{2}}{2}$
$\sum_{n=0}^{\infty} \frac{1}{x_{n}^{3}} = -{\gamma}^{3} - \frac{\gamma {\pi}^{2}}{2} - 4 \zeta\!\left(3\right)$
$\sum_{n=0}^{\infty} \frac{1}{x_{n}^{4}} = {\gamma}^{4} + \frac{{\pi}^{4}}{9} + \frac{2 {\gamma}^{2} {\pi}^{2}}{3} + 4 \gamma \zeta\!\left(3\right)$
$\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC