# Digamma function

## Definitions

Symbol: DigammaFunction $\psi\!\left(z\right)$ Digamma function
Symbol: DigammaFunctionZero $x_{n}$ Zero of the digamma function

## Illustrations

Image: X-ray of $\psi\!\left(z\right)$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$
Image: X-ray of $\psi'\!\left(z\right)$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$

## Domain and singularities

### Digamma function

$x \in \mathbb{R} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(x\right) \in \mathbb{R}$
$z \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(z\right) \in \mathbb{C}$
$\psi\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}$
$\psi\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}$
$\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \{0, -1, \ldots\}$

### Polygamma functions

$\psi^{(0)}\!\left(z\right) = \psi\!\left(z\right)$
$\left(m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \implies \left(\psi^{(m)}\!\left(x\right) \in \mathbb{R}\right)$
$\left(m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\psi^{(m)}\!\left(z\right) \in \mathbb{C}\right)$
$\psi^{(m)}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}$
$\psi^{(m)}\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}$
$\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi^{(m)}\!\left(z\right) = \{0, -1, \ldots\}$

## Specific values

Main topic: Specific values of the 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 and simple fractions

$\psi\!\left(1\right) = -\gamma$
$\psi\!\left(2\right) = 1 - \gamma$
$\psi\!\left(n\right) = H_{n - 1} - \gamma$
$\psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma$
$\psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2}$
$\psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2)$
$\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)$

### Values of polygamma functions at integers and simple fractions

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

## Zeros

$n \in \mathbb{Z}_{\ge 0} \;\implies\; x_{n} \in \mathbb{R}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \left\{ x_{n} : n \in \mathbb{Z}_{\ge 0} \right\}$
$x_{n} = \mathop{\operatorname{zero*}\,}\limits_{x \in S} \psi\!\left(x\right)\; \text{ where } S = \begin{cases} \left(0, \infty\right), & n = 0\\\left(-n, -n + 1\right), & n < 0\\ \end{cases}$
$x_{n} \sim -n + \frac{1}{\pi} \operatorname{atan}\!\left(\frac{\pi}{\log(n)}\right), \; n \to \infty$

## Derivatives and differential equations

Related topic: Gamma function
$\psi\!\left(z\right) = \frac{\Gamma'(z)}{\Gamma(z)}$
$\psi\!\left(z\right) = \frac{d}{d z}\, \left[\log \Gamma(z)\right]$
$\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]$
$\frac{d^{n}}{{d z}^{n}} \left[\psi\!\left(z\right)\right] = \psi^{(n)}\!\left(z\right)$
$\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)$

## Series representations

### Series of rational functions

$\psi\!\left(z\right) = -\gamma + \sum_{n=0}^{\infty} \left(\frac{1}{n + 1} - \frac{1}{n + z}\right)$
$\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \sum_{n=0}^{\infty} \frac{1}{{\left(n + z\right)}^{m + 1}}$

### Taylor series

$\psi\!\left(1 + z\right) = -\gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}$

### Laurent series

$\psi\!\left(z\right) = -\frac{1}{z} - \gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}$
$\psi\!\left(-n + z\right) = -\frac{1}{z} + \psi\!\left(n + 1\right) + \sum_{k=1}^{\infty} \left({\left(-1\right)}^{k + 1} \zeta\!\left(k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + 1}}\right) {z}^{k}$
$\psi^{(m)}\!\left(-n + z\right) = \frac{{\left(-1\right)}^{m + 1} m !}{{z}^{m + 1}} + \sum_{k=0}^{\infty} \left(k + 1\right)_{m} \left({\left(-1\right)}^{m + k + 1} \zeta\!\left(m + k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + m + 1}}\right) {z}^{k}$

### Asymptotic expansions

$\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)$
$\psi^{(m)}\!\left(z\right) = \frac{{\left(-1\right)}^{m + 1}}{m !} \left(\frac{1}{m {z}^{m}} + \frac{1}{2 {z}^{m + 1}} + \sum_{n=1}^{N - 1} \frac{\left(m + 1\right)_{2 n - 1}}{\left(2 n\right)!} \frac{B_{2 n}}{{z}^{m + 2 n}}\right) + {R}^{(m + 1)}_{N}(z)$

### Weierstrass product

$\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)$

## Representation by other functions

$\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)$
$\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \Phi\!\left(1, m + 1, z\right)$
$\psi\!\left(z\right) = \left(z - 1\right) \,{}_3F_2\!\left(1, 1, 2 - z, 2, 2, 1\right) - \gamma$
$\psi\!\left(z\right) = -\gamma_{0}\!\left(z\right)$

## Functional equations

### Recurrence relations

$\psi\!\left(z + 1\right) = \psi\!\left(z\right) + \frac{1}{z}$
$\psi\!\left(z + n\right) = \psi\!\left(z\right) + \sum_{k=0}^{n - 1} \frac{1}{z + k}$
$\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}$
$\psi^{(m)}\!\left(z + 1\right) = \psi^{(m)}\!\left(z\right) + \frac{{\left(-1\right)}^{m} m !}{{z}^{m + 1}}$
$\psi^{(m)}\!\left(z + n\right) = \psi^{(m)}\!\left(z\right) + {\left(-1\right)}^{m} m ! \sum_{k=0}^{n - 1} \frac{1}{{\left(z + k\right)}^{m + 1}}$
$\psi^{(m)}\!\left(z - n\right) = \psi^{(m)}\!\left(z\right) - {\left(-1\right)}^{m} m ! \sum_{k=1}^{n} \frac{1}{{\left(z - k\right)}^{m + 1}}$

### Reflection formula

$\psi\!\left(1 - z\right) = \psi\!\left(z\right) + \pi \cot\!\left(\pi z\right)$
$\psi^{(m)}\!\left(1 - z\right) = {\left(-1\right)}^{m} \left(\psi^{(m)}\!\left(z\right) + \pi \frac{d^{m}}{{d z}^{m}} \cot\!\left(\pi z\right)\right)$

### Multiplication theorem

$\psi\!\left(n z\right) = \log(n) + \frac{1}{n} \sum_{k=0}^{n - 1} \psi\!\left(z + \frac{k}{n}\right)$
$\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)$

### Conjugate symmetry

$\psi\!\left(\overline{z}\right) = \overline{\psi\!\left(z\right)}$
$\psi^{(m)}\!\left(\overline{z}\right) = \overline{\psi^{(m)}\!\left(z\right)}$

## Integral representations

$\psi\!\left(z\right) = \int_{0}^{\infty} \left(\frac{{e}^{-t}}{t} - \frac{{e}^{-z t}}{1 - {e}^{-t}}\right) \, dt$
$\psi\!\left(z\right) = \int_{0}^{\infty} \left({e}^{-t} - \frac{1}{{\left(1 + t\right)}^{z}}\right) \frac{1}{t} \, dt$
$\psi\!\left(z\right) = -\gamma + \int_{0}^{1} \frac{1 - {t}^{z - 1}}{1 - t} \, dt$
$\psi\!\left(z\right) = -\gamma + \int_{0}^{\infty} \frac{{e}^{-t} - {e}^{-z t}}{1 - {e}^{-t}} \, dt$
$\psi\!\left(z\right) = \log(z) + \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{t} - \frac{1}{1 - {e}^{-t}}\right) \, dt$
$\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - 2 \int_{0}^{\infty} \frac{t}{\left({t}^{2} + {z}^{2}\right) \left({e}^{2 \pi t} - 1\right)} \, dt$
$\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt$
$\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, dt$
$\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt$

## Generating functions

$\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}$
$\sum_{n=1}^{\infty} \psi\!\left(n\right) \frac{{z}^{n}}{n !} = z \left[ \frac{d}{d a}\, \,{}_1F_1\!\left(a, 2, z\right) \right]_{a = 1} - \gamma \left({e}^{z} - 1\right)$

## Finite sums

$\sum_{k=1}^{n} \psi\!\left(k\right) = n \left(\psi\!\left(n + 1\right) - 1\right)$
$\sum_{k=1}^{n} \psi\!\left(k\right) {e}^{2 \pi r k i / n} = n \log\!\left(1 - {e}^{2 \pi r i / n}\right)$
$\sum_{k=1}^{n} \psi\!\left(k\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)$