Fungrim entry: 45267a

\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{3}}

References:

https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty

TeX:

\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta\!\left(3\right)}{{\pi}^{3}}

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
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("45267a"),
    Formula(Equal(Integral(Div(Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 8), Add(1, Pow(t, 2))), For(t, 0, Infinity)), Div(Mul(20, RiemannZeta(3)), Pow(Pi, 3)))),
    References("https://math.stackexchange.com/questions/1760270/closed-form-of-an-integral-involving-a-jacobi-theta-function-int-0-infty"))

Topics using this entry

Integrals of Jacobi theta functions