# Fungrim entry: 361f61

$\psi^{(m)}\!\left(1 - z\right) = {\left(-1\right)}^{m} \left(\psi^{(m)}\!\left(z\right) + \pi \frac{d^{m}}{{d z}^{m}} \cot\!\left(\pi z\right)\right)$
Assumptions:$m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}$
TeX:
\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)

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("361f61"),
Formula(Equal(DigammaFunction(Sub(1, z), m), Mul(Pow(-1, m), Add(DigammaFunction(z, m), Mul(Pi, ComplexDerivative(Cot(Mul(Pi, z)), For(z, z, m))))))),
Variables(m, z),
Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZ))))

## 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