Fungrim entry: 1699a9

\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:

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}

References:

https://doi.org/10.1090/mcom/3401

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("1699a9"),
    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)))),
    References("https://doi.org/10.1090/mcom/3401"))

Topics using this entry

Hurwitz zeta function