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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}
\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}
Fungrim symbol Notation Short description
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\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:
    Formula(Equal(Zeros(JacobiTheta(3, 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))))),
    Assumptions(Element(tau, HH)))

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2019-08-19 14:38:23.809000 UTC