# Fungrim entry: 547fcd

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Log$\log(z)$ Natural logarithm
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("547fcd"),
Formula(Equal(DigammaFunction(z, m), Neg(Integral(Mul(Div(Pow(t, Sub(z, 1)), Sub(1, t)), Pow(Log(t), m)), For(t, 0, 1))))),
Variables(z, m),
Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(m, ZZGreaterEqual(1)))))

## 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