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Fungrim entry: 2429b2

∫M/2N/2θ3 ⁣(x,τ) dx=N−M2\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}
Assumptions:τ∈H  and⁡  M∈Z  and⁡  N∈Z\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}
TeX:
\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Integral∫abf(x) dx\int_{a}^{b} f(x) \, dx Integral
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
HHH\mathbb{H} Upper complex half-plane
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("2429b2"),
    Formula(Equal(Integral(JacobiTheta(3, x, tau), For(x, Div(M, 2), Div(N, 2))), Div(Sub(N, M), 2))),
    Variables(tau, M, N),
    Assumptions(And(Element(tau, HH), Element(M, ZZ), Element(N, ZZ))))

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

2021-03-15 19:12:00.328586 UTC