Fungrim entry: 1a15f9

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

Source code for this entry:

Entry(ID("1a15f9"),
    Formula(Equal(Integral(Div(Mul(Pow(JacobiTheta(2, 0, Mul(ConstI, t)), 4), Pow(JacobiTheta(4, 0, Mul(ConstI, t)), 2)), Add(1, Pow(t, 2))), For(t, 0, Infinity)), Div(2, 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