Fungrim entry: 498036

\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

TeX:

\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Gamma`	$\Gamma(z)$	Gamma function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("498036"),
    Formula(Equal(HurwitzZeta(s, a), Mul(Div(1, Gamma(s)), Integral(Div(Mul(Pow(x, Sub(s, 1)), Exp(Neg(Mul(a, x)))), Sub(1, Exp(Neg(x)))), For(x, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, CC), Greater(Re(a), 0))))

Topics using this entry

Hurwitz zeta function