Fungrim home page

Fungrim entry: 60c6da

ζ ⁣(s,a)=1s1+n=0(1)nn!γn ⁣(a)(s1)n\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}
Assumptions:sC  and  aC  and  a{0,1,}s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
TeX:
\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
StieltjesGammaγn ⁣(a)\gamma_{n}\!\left(a\right) Stieltjes constant
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("60c6da"),
    Formula(Equal(HurwitzZeta(s, a), Add(Div(1, Sub(s, 1)), Sum(Mul(Mul(Div(Pow(-1, n), Factorial(n)), StieltjesGamma(n, a)), Pow(Sub(s, 1), n)), For(n, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

Topics using this entry

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

2020-08-27 09:56:25.682319 UTC