Fungrim entry: eec21a

$\psi\!\left(n z\right) = \log(n) + \frac{1}{n} \sum_{k=0}^{n - 1} \psi\!\left(z + \frac{k}{n}\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}$
TeX:
\psi\!\left(n z\right) = \log(n) + \frac{1}{n} \sum_{k=0}^{n - 1} \psi\!\left(z + \frac{k}{n}\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Log$\log(z)$ Natural logarithm
Sum$\sum_{n} f(n)$ Sum
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("eec21a"),
Formula(Equal(DigammaFunction(Mul(n, z)), Add(Log(n), Mul(Div(1, n), Sum(DigammaFunction(Add(z, Div(k, n))), For(k, 0, Sub(n, 1))))))),
Variables(n, z),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC), NotElement(Mul(n, z), ZZLessEqual(0)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC