# Fungrim entry: d9c818

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

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
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("d9c818"),
Formula(Equal(DigammaFunction(z), Sub(Sub(Log(z), Div(1, Mul(2, z))), Mul(2, Integral(Div(t, Mul(Add(Pow(t, 2), Pow(z, 2)), Sub(Exp(Mul(Mul(2, Pi), 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