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Fungrim entry: f79ff0

zexzez1=n=0Bn ⁣(x)znn!\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}
Assumptions:xCandzCandz<2πandz0x \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
Expez{e}^{z} Exponential function
Sumnf(n)\sum_{n} f(n) Sum
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ConstPiπ\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, ConstPi)), Unequal(z, 0))))

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

2019-10-05 13:11:19.856591 UTC