# Fungrim entry: bf533f

$\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}$
References:
• https://doi.org/10.1080%2F10652469.2017.1376193
TeX:
\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Gamma$\Gamma(z)$ Gamma function
Exp${e}^{z}$ Exponential function
ConstGamma$\gamma$ The constant gamma (0.577...)
Product$\prod_{n} f(n)$ Product
DigammaFunctionZero$x_{n}$ Zero of the digamma function
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("bf533f"),
Formula(Equal(Div(DigammaFunction(z), Gamma(z)), Neg(Mul(Exp(Mul(Mul(2, ConstGamma), z)), Product(Mul(Sub(1, Div(z, DigammaFunctionZero(n))), Exp(Div(z, DigammaFunctionZero(n)))), For(n, 0, Infinity)))))),
Variables(z),
Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))),
References("https://doi.org/10.1080%2F10652469.2017.1376193"))

## 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