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Fungrim entry: 1699a9

ζ ⁣(s,a)=π2(s1)(a12+ix)1scosh2 ⁣(πx)dx\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx
Assumptions:sC  and  s1  and  aC  and  Re(a)>12s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}
\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Piπ\pi The constant pi (3.14...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
ConstIii Imaginary unit
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(Pi, Mul(2, Sub(s, 1))), Integral(Div(Pow(Add(Sub(a, Div(1, 2)), Mul(ConstI, x)), Sub(1, s)), Pow(Cosh(Mul(Pi, x)), 2)), For(x, Neg(Infinity), Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), Div(1, 2)))),

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2021-03-15 19:12:00.328586 UTC