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Fungrim entry: 140815

0(θ4 ⁣(0,it)1)2dt=π3log ⁣(2)\int_{0}^{\infty} {\left(\theta_{4}\!\left(0 , i t\right) - 1\right)}^{2} \, dt = \frac{\pi}{3} - \log\!\left(2\right)
TeX:
\int_{0}^{\infty} {\left(\theta_{4}\!\left(0 , i t\right) - 1\right)}^{2} \, dt = \frac{\pi}{3} - \log\!\left(2\right)
Definitions:
Fungrim symbol Notation Short description
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Infinity\infty Positive infinity
ConstPiπ\pi The constant pi (3.14...)
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Source code for this entry:
Entry(ID("140815"),
    Formula(Equal(Integral(Pow(Sub(JacobiTheta(4, 0, Mul(ConstI, t)), 1), 2), Tuple(t, 0, Infinity)), Sub(Div(ConstPi, 3), Log(2)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-22 15:43:45.410764 UTC