# Fungrim entry: 4f5575

$\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0$
TeX:
\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Log$\log(z)$ Natural logarithm
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Exp${e}^{z}$ Exponential function
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("4f5575"),
Formula(Equal(DigammaFunction(z), Sub(Sub(Log(z), Div(1, Mul(2, z))), Integral(Mul(Exp(Neg(Mul(z, t))), Parentheses(Add(Sub(Div(1, 2), Div(1, t)), Div(1, Sub(Exp(t), 1))))), For(t, 0, Infinity))))),
Variables(z),
Assumptions(And(Element(z, CC), Greater(Re(z), 0))))

## Topics using this entry

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