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Jacobi theta functions

Table of contents: Exponential Fourier series - Trigonometric Fourier series - Zeros - Symmetry - Periodicity - Quasi-periodicity

Exponential Fourier series

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θ1 ⁣(z,τ)=in=(1)neπi((n+1/2)2τ+(2n+1)z)\theta_1\!\left(z, \tau\right) = -i \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}
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θ2 ⁣(z,τ)=n=eπi((n+1/2)2τ+(2n+1)z)\theta_2\!\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\right)}
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θ3 ⁣(z,τ)=n=eπi(n2τ+2nz)\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}
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θ4 ⁣(z,τ)=n=(1)neπi(n2τ+2nz)\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

Trigonometric Fourier series

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θ1 ⁣(z,τ)=2n=0(1)neπi(n+1/2)2τsin ⁣((2n+1)πz)\theta_1\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \sin\!\left(\left(2 n + 1\right) \pi z\right)
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θ2 ⁣(z,τ)=2n=0eπi(n+1/2)2τcos ⁣((2n+1)πz)\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)
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θ3 ⁣(z,τ)=1+2n=1eπin2τcos ⁣(2nπz)\theta_3\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)
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θ4 ⁣(z,τ)=1+2n=1(1)neπin2τcos ⁣(2nπz)\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

Zeros

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zeroszCθ1 ⁣(z,τ)={m+nτ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_1\!\left(z, \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
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zeroszCθ2 ⁣(z,τ)={(m+12)+nτ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_2\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
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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\}
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zeroszCθ4 ⁣(z,τ)={m+(n+12)τ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_4\!\left(z, \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Symmetry

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θ1 ⁣(z,τ)=θ1 ⁣(z,τ)\theta_1\!\left(-z, \tau\right) = -\theta_1\!\left(z, \tau\right)
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θ2 ⁣(z,τ)=θ2 ⁣(z,τ)\theta_2\!\left(-z, \tau\right) = \theta_2\!\left(z, \tau\right)
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θ3 ⁣(z,τ)=θ3 ⁣(z,τ)\theta_3\!\left(-z, \tau\right) = \theta_3\!\left(z, \tau\right)
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θ4 ⁣(z,τ)=θ4 ⁣(z,τ)\theta_4\!\left(-z, \tau\right) = \theta_4\!\left(z, \tau\right)

Periodicity

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θ1 ⁣(z+2n,τ)=θ1 ⁣(z,τ)\theta_1\!\left(z + 2 n, \tau\right) = \theta_1\!\left(z, \tau\right)
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θ2 ⁣(z+2n,τ)=θ2 ⁣(z,τ)\theta_2\!\left(z + 2 n, \tau\right) = \theta_2\!\left(z, \tau\right)
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θ3 ⁣(z+2n,τ)=θ3 ⁣(z,τ)\theta_3\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)
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θ4 ⁣(z+2n,τ)=θ3 ⁣(z,τ)\theta_4\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)

Quasi-periodicity

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θ1 ⁣(z+m+nτ,τ)=(1)m+neπi(τn2+2nz)θ1 ⁣(z,τ)\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)
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θ2 ⁣(z+m+nτ,τ)=(1)meπi(τn2+2nz)θ2 ⁣(z,τ)\theta_2\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_2\!\left(z, \tau\right)
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θ3 ⁣(z+m+nτ,τ)=eπi(τn2+2nz)θ3 ⁣(z,τ)\theta_3\!\left(z + m + n \tau, \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_3\!\left(z, \tau\right)
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θ4 ⁣(z+m+nτ,τ)=(1)neπi(τn2+2nz)θ4 ⁣(z,τ)\theta_4\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_4\!\left(z, \tau\right)

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

2019-05-23 08:00:13.607731 UTC