# Fungrim entry: 15d56a

$\left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \begin{cases} \left(0, \infty\right), & m \text{ odd}\\\left(-\infty, 0\right), & m \text{ even}\\ \end{cases}$
Assumptions:$m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}$
TeX:
\left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \begin{cases} \left(0, \infty\right), & m \text{ odd}\\\left(-\infty, 0\right), & m \text{ even}\\ \end{cases}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DigammaFunction$\psi\!\left(z\right)$ Digamma function
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("15d56a"),
Formula(Implies(And(Element(x, OpenInterval(0, Infinity)), Element(m, ZZGreaterEqual(1))), Element(DigammaFunction(x, m), Cases(Tuple(OpenInterval(0, Infinity), Odd(m)), Tuple(OpenInterval(Neg(Infinity), 0), Even(m)))))),
Variables(x, m),
Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(x, RR))))

## 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