Fungrim entry: 41631f

\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{3}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2

TeX:

\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{3}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2

Definitions:

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

Source code for this entry:

Entry(ID("41631f"),
    Formula(Equal(Integral(Mul(Pow(t, Sub(s, 1)), Sub(JacobiTheta(3, 0, Mul(ConstI, Pow(t, 2))), 1)), For(t, 0, Infinity)), Mul(Mul(Pow(Pi, Neg(Div(s, 2))), Gamma(Div(s, 2))), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), Greater(Re(s), 2))))

Fungrim entry: 41631f

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