# Fungrim entry: 4e3853

$\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1$
TeX:
\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
ConstGamma$\gamma$ The constant gamma (0.577...)
Log$\log(z)$ Natural logarithm
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("4e3853"),
Formula(Equal(Sum(Mul(DigammaFunction(n), Pow(z, n)), For(n, 1, Infinity)), Div(Mul(z, Add(ConstGamma, Log(Sub(1, z)))), Sub(z, 1)))),
Variables(z),
Assumptions(And(Element(z, CC), Less(Abs(z), 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