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Fungrim entry: 9376ec

0ts1θ2 ⁣(0,it2)dt=(2s1)πs/2Γ ⁣(s2)ζ ⁣(s)\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)
Assumptions:sC  and  Re(s)>2s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2
\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, 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
Piπ\pi The constant pi (3.14...)
GammaΓ(z)\Gamma(z) Gamma function
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Equal(Integral(Mul(Pow(t, Sub(s, 1)), JacobiTheta(2, 0, Mul(ConstI, Pow(t, 2)))), For(t, 0, Infinity)), Mul(Mul(Mul(Sub(Pow(2, s), 1), Pow(Pi, Neg(Div(s, 2)))), Gamma(Div(s, 2))), RiemannZeta(s)))),
    Assumptions(And(Element(s, CC), Greater(Re(s), 2))))

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2021-03-15 19:12:00.328586 UTC