Fungrim entry: e4cdf1

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

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}^{4}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{4 \log(2)}{\pi}

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
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("e4cdf1"),
    Formula(Equal(Integral(Div(Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 4), Add(1, Pow(t, 2))), For(t, 0, Infinity)), Div(Mul(4, Log(2)), Pi))),
    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