# Fungrim entry: f79ff0

$\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0$
TeX:
\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0
Definitions:
Fungrim symbol Notation Short description
Exp${e}^{z}$ Exponential function
Sum$\sum_{n} f(n)$ Sum
BernoulliPolynomial$B_{n}\!\left(z\right)$ Bernoulli polynomial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Pi$\pi$ The constant pi (3.14...)
Source code for this entry:
Entry(ID("f79ff0"),
Formula(Equal(Div(Mul(z, Exp(Mul(x, z))), Sub(Exp(z), 1)), Sum(Mul(BernoulliPolynomial(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
Variables(z, x),
Assumptions(And(Element(x, CC), Element(z, CC), Less(Abs(z), Mul(2, Pi)), NotEqual(z, 0))))

## 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