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Fungrim entry: caf10a

zeroszCθ3 ⁣(z,τ)={(m+12)+(n+12)τ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_3\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
Assumptions:τH\tau \in \mathbb{H}
TeX:
\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_3\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta3θ3 ⁣(z,τ)\theta_3\!\left(z, \tau\right) Jacobi theta function
CCC\mathbb{C} Complex numbers
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZZ\mathbb{Z} Integers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("caf10a"),
    Formula(Equal(Zeros(JacobiTheta3(z, tau), z, Element(z, CC)), SetBuilder(Add(Parentheses(Add(m, Div(1, 2))), Mul(Add(n, Div(1, 2)), tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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

2019-06-18 07:49:59.356594 UTC