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Fungrim entry: d637c5

θj ⁣(z,τ+x)=n=01(4πi)nθj(2n) ⁣(z,τ)n!xn\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}
Assumptions:j{1,2,3,4}  and  zC  and  τH  and  xC  and  x<Im(τ)j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)
\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Absz\left|z\right| Absolute value
ImIm(z)\operatorname{Im}(z) Imaginary part
Source code for this entry:
    Formula(Equal(JacobiTheta(j, z, Add(tau, x)), Sum(Mul(Mul(Div(1, Pow(Mul(Mul(4, Pi), ConstI), n)), Div(JacobiTheta(j, z, tau, Mul(2, n)), Factorial(n))), Pow(x, n)), For(n, 0, Infinity)))),
    Variables(j, z, tau, x),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(x, CC), Less(Abs(x), Im(tau)))))

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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