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Fungrim entry: ed4ce5

θ1 ⁣(z,τ)=n=eπi((n+1/2)2τ+(2n+1)z+n1/2)\theta_{1}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z + n - 1 / 2\right)}
Assumptions:zCandτHz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{1}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z + n - 1 / 2\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf(n)\sum_{n} f(n) Sum
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("ed4ce5"),
    Formula(Equal(JacobiTheta(1, z, tau), Sum(Exp(Mul(Mul(ConstPi, ConstI), Sub(Add(Add(Mul(Pow(Add(n, Div(1, 2)), 2), tau), Mul(Add(Mul(2, n), 1), z)), n), Div(1, 2)))), For(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2019-10-05 13:11:19.856591 UTC