Fungrim entry: 522b04

\frac{z}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n} \frac{{z}^{n}}{n !}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt 2 \pi \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\frac{z}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n} \frac{{z}^{n}}{n !}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt 2 \pi \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("522b04"),
    Formula(Equal(Div(z, Sub(Exp(z), 1)), Sum(Mul(BernoulliB(n), Div(Pow(z, n), Factorial(n))), Tuple(n, 0, Infinity)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), Mul(2, ConstPi)), Unequal(z, 0))))

Topics using this entry

Bernoulli numbers and polynomials