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Fungrim entry: 498036

ζ ⁣(s,a)=1Γ(s)0xs1eax1exdx\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx
Assumptions:sC  and  Re(s)>1  and  aC  and  Re(a)>0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0
\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
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
GammaΓ(z)\Gamma(z) Gamma function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    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))))

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2021-03-15 19:12:00.328586 UTC