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Jacobi theta functions

Table of contents: Definitions - Illustrations - Series and product representations - Zeros - Specific values - Argument transformations - Lattice transformations - Sums and products - Differential equations - Integrals - Representation of other functions - Representation by other functions - Approximations

Definitions

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Symbol: JacobiTheta — θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) — Jacobi theta function

Illustrations

Main topic: Illustrations of Jacobi theta functions

Variable argument

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Image: X-ray of θ3 ⁣(z,i)\theta_{3}\!\left(z , i\right) on z∈[−2,2]+[−2,2]iz \in \left[-2, 2\right] + \left[-2, 2\right] i

Variable lattice parameter

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Image: X-ray of θ1 ⁣(13+34i,τ)\theta_{1}\!\left(\frac{1}{3} + \frac{3}{4} i , \tau\right) on τ∈[−52,52]+[0,2]i\tau \in \left[-\frac{5}{2}, \frac{5}{2}\right] + \left[0, 2\right] i

Series and product representations

Main topic: Series and product representations of Jacobi theta functions

Trigonometric Fourier series

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θ1 ⁣(z,τ)=2eπiτ/4∑n=0∞(−1)nqn(n+1)sin⁡ ⁣((2n+1)πz)   where q=eπiτ\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
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θ2 ⁣(z,τ)=2eπiτ/4∑n=0∞qn(n+1)cos⁡ ⁣((2n+1)πz)   where q=eπiτ\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
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θ3 ⁣(z,τ)=1+2∑n=1∞qn2cos⁡ ⁣(2nπz)   where q=eπiτ\theta_{3}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
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θ4 ⁣(z,τ)=1+2∑n=1∞(−1)nqn2cos⁡ ⁣(2nπz)   where q=eπiτ\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Jacobi triple product

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θ3 ⁣(z,τ)=∑n=−∞∞qn2w2n=∏n=1∞(1−q2n)(1+q2n−1w2)(1+q2n−1w−2)   where q=eπiτ,  w=eπiz\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Zeros

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zeros⁡ z∈Cθ1 ⁣(z,τ)={m+nτ:m∈Z  and⁡  n∈Z}\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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zeros⁡ z∈Cθ2 ⁣(z,τ)={(m+12)+nτ:m∈Z  and⁡  n∈Z}\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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zeros⁡ z∈Cθ3 ⁣(z,τ)={(m+12)+(n+12)τ:m∈Z  and⁡  n∈Z}\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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zeros⁡ z∈Cθ4 ⁣(z,τ)={m+(n+12)τ:m∈Z  and⁡  n∈Z}\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\}

Specific values

Main topic: Specific values of Jacobi theta functions
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θ1 ⁣(0,τ)=0\theta_{1}\!\left(0 , \tau\right) = 0
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θ3 ⁣(0,i)=π1/4Γ ⁣(34)\theta_{3}\!\left(0 , i\right) = \frac{{\pi}^{1 / 4}}{\Gamma\!\left(\frac{3}{4}\right)}
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θ2 ⁣(0,i)=θ4 ⁣(0,i)=[2−1/4]θ3 ⁣(0,i)\theta_{2}\!\left(0 , i\right) = \theta_{4}\!\left(0 , i\right) = \left[{2}^{-1 / 4}\right] \theta_{3}\!\left(0 , i\right)

Argument transformations

Main topic: Argument transformations for Jacobi theta functions

Even-odd 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)

Conjugate symmetry

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θj ⁣(z‾,τ)=θj ⁣(z,−τ‾)‾\theta_{j}\!\left(\overline{z} , \tau\right) = \overline{\theta_{j}\!\left(z , -\overline{\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+n,τ)=θ3 ⁣(z,τ)\theta_{3}\!\left(z + n , \tau\right) = \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z+n,τ)=θ4 ⁣(z,τ)\theta_{4}\!\left(z + n , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Quasi-periodicity

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θ1 ⁣(z+m+nτ,τ)=(−1)m+ne−πi(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)me−πi(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)=e−πi(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)ne−πi(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)

Lattice transformations

Main topic: Lattice transformations for Jacobi theta functions

Basic modular transformations

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θ1 ⁣(z,τ+1)=eπi/4θ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z,τ+1)=eπi/4θ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z,τ+1)=θ4 ⁣(z,τ)\theta_{3}\!\left(z , \tau + 1\right) = \theta_{4}\!\left(z , \tau\right)
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θ4 ⁣(z,τ+1)=θ3 ⁣(z,τ)\theta_{4}\!\left(z , \tau + 1\right) = \theta_{3}\!\left(z , \tau\right)
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θ1 ⁣(z,−1τ)=−iτieπiτz2θ1 ⁣(τz,τ)\theta_{1}\!\left(z , -\frac{1}{\tau}\right) = -i \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{1}\!\left(\tau z , \tau\right)
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θ2 ⁣(z,−1τ)=τieπiτz2θ4 ⁣(τz,τ)\theta_{2}\!\left(z , -\frac{1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{4}\!\left(\tau z , \tau\right)
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θ3 ⁣(z,−1τ)=τieπiτz2θ3 ⁣(τz,τ)\theta_{3}\!\left(z , -\frac{1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{3}\!\left(\tau z , \tau\right)
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θ4 ⁣(z,−1τ)=τieπiτz2θ2 ⁣(τz,τ)\theta_{4}\!\left(z , -\frac{1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{2}\!\left(\tau z , \tau\right)

General modular transformations

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θj ⁣(z,aτ+bcτ+d)=εj ⁣(a,b,c,d)vieπicvz2θSj ⁣(a,b,c,d) ⁣(vz,τ)   where v=cτ+d\theta_{j}\!\left(z , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d

Multiplication of the lattice parameter

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θ3 ⁣(z,τ2)=θ22 ⁣(z,τ)+θ32 ⁣(z,τ)θ3 ⁣(0,τ2)\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \frac{\tau}{2}\right)}
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θ3 ⁣(2z,2τ)=θ12 ⁣(z,τ)+θ22 ⁣(z,τ)2θ2 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right)}{2 \theta_{2}\!\left(0 , 2 \tau\right)}
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θ3 ⁣(2z,4τ)=θ3 ⁣(z,τ)+θ4 ⁣(z,τ)2\theta_{3}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) + \theta_{4}\!\left(z , \tau\right)}{2}

Sums and products

See also Argument transformations for Jacobi theta functions and Lattice transformations for Jacobi theta functions for sum and product identities involving transformations.

Fourth powers

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θ34 ⁣(0,τ)=θ24 ⁣(0,τ)+θ44 ⁣(0,τ)\theta_{3}^{4}\!\left(0, \tau\right) = \theta_{2}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)
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θ14 ⁣(z,τ)+θ34 ⁣(z,τ)=θ24 ⁣(z,τ)+θ44 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) + \theta_{3}^{4}\!\left(z, \tau\right) = \theta_{2}^{4}\!\left(z, \tau\right) + \theta_{4}^{4}\!\left(z, \tau\right)
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θ14 ⁣(z,τ)−θ24 ⁣(z,τ)=θ44 ⁣(z,τ)−θ34 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) - \theta_{2}^{4}\!\left(z, \tau\right) = \theta_{4}^{4}\!\left(z, \tau\right) - \theta_{3}^{4}\!\left(z, \tau\right)
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θ14 ⁣(z,τ)−θ44 ⁣(z,τ)=θ24 ⁣(z,τ)−θ34 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) - \theta_{4}^{4}\!\left(z, \tau\right) = \theta_{2}^{4}\!\left(z, \tau\right) - \theta_{3}^{4}\!\left(z, \tau\right)

Sums of squares

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θ22 ⁣(0,τ)θ32 ⁣(z,τ)=θ42 ⁣(0,τ)θ12 ⁣(z,τ)+θ32 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{2}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) = \theta_{4}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
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θ22 ⁣(0,τ)θ42 ⁣(z,τ)=θ32 ⁣(0,τ)θ12 ⁣(z,τ)+θ42 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{2}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
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θ32 ⁣(0,τ)θ22 ⁣(z,τ)=θ22 ⁣(0,τ)θ32 ⁣(z,τ)−θ42 ⁣(0,τ)θ12 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) - \theta_{4}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right)
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θ32 ⁣(0,τ)θ32 ⁣(z,τ)=θ42 ⁣(0,τ)θ42 ⁣(z,τ)+θ22 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) = \theta_{4}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
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θ32 ⁣(0,τ)θ42 ⁣(z,τ)=θ22 ⁣(0,τ)θ12 ⁣(z,τ)+θ42 ⁣(0,τ)θ32 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right)
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θ2 ⁣(0,2τ)(θ12 ⁣(z,τ)−θ22 ⁣(z,τ))=θ3 ⁣(0,2τ)(θ42 ⁣(z,τ)−θ32 ⁣(z,τ))\theta_{2}\!\left(0 , 2 \tau\right) \left(\theta_{1}^{2}\!\left(z, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right)\right) = \theta_{3}\!\left(0 , 2 \tau\right) \left(\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right)\right)

Differential equations

Main topic: Differential equations for Jacobi theta functions

Notation and conversion to argument derivatives

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drdzrθj ⁣(z,τ)=θj(r) ⁣(z,τ)\frac{d^{r}}{{d z}^{r}} \theta_{j}\!\left(z , \tau\right) = \theta^{(r)}_{j}\!\left(z , \tau\right)
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drdτrθj(s) ⁣(z,τ)=1(4πi)rθj(2r+s) ⁣(z,τ)\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)

Heat equation

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θj′′ ⁣(z,τ)−4πiddτ θj ⁣(z,τ)=0\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

Jacobi's differential equation

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(30D13−15D0D1D2+D02D3)2+32(D0D2−3D12)3+π2(D0D2−3D12)2D010=0   where Dr=drdτrθj ⁣(0,τ){\left(30 D_{1}^{3} - 15 D_{0} D_{1} D_{2} + D_{0}^{2} D_{3}\right)}^{2} + 32 {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{3} + {\pi}^{2} {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{2} D_{0}^{10} = 0\; \text{ where } D_{r} = \frac{d^{r}}{{d \tau}^{r}} \theta_{j}\!\left(0 , \tau\right)

Derivatives of ratios

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ddz θ1 ⁣(z,τ)θ2 ⁣(z,τ)=πθ22 ⁣(0,τ)θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ22 ⁣(z,τ)\frac{d}{d z}\, \frac{\theta_{1}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}^{2}\!\left(z, \tau\right)}

Integrals

Main topic: Integrals of Jacobi theta functions

Laplace transforms

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∫0∞e−atθ1 ⁣(x,ibt) dt=πabsinh⁡ ⁣(2xπab)cosh⁡ ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
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∫0∞e−atθ2 ⁣(x,ibt) dt=−πabsinh⁡ ⁣((2x−1)πab)cosh⁡ ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(x , i b t\right) \, dt = -\sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
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∫0∞e−atθ3 ⁣(x,ibt) dt=πabcosh⁡ ⁣((2x−1)πab)sinh⁡ ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
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∫0∞e−atθ4 ⁣(x,ibt) dt=πabcosh⁡ ⁣(2xπab)sinh⁡ ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

Mellin transforms

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∫0∞ts−1θ2 ⁣(0,it2) dt=(2s−1)π−s/2Γ ⁣(s2)ζ ⁣(s)\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)
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∫0∞ts−1(θ3 ⁣(0,it2)−1) dt=π−s/2Γ ⁣(s2)ζ ⁣(s)\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{3}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)
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∫0∞ts−1(θ4 ⁣(0,it2)−1) dt=(21−s−1)π−s/2Γ ⁣(s2)ζ ⁣(s)\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{4}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = \left({2}^{1 - s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta\!\left(s\right)

Constant definite integrals

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∫0∞θ2 ⁣(0,it) dt=π\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \, dt = \pi
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∫0∞θ2 ⁣(0,it)θ3 ⁣(0,it)θ4 ⁣(0,it) dt=2\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \theta_{3}\!\left(0 , i t\right) \theta_{4}\!\left(0 , i t\right) \, dt = 2

Representation of other functions

Modular forms and functions

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η(τ)=eπiτ/12θ3 ⁣(τ+12,3τ)\eta(\tau) = {e}^{\pi i \tau / 12} \theta_{3}\!\left(\frac{\tau + 1}{2} , 3 \tau\right)
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j(τ)=32(θ28 ⁣(0,τ)+θ38 ⁣(0,τ)+θ48 ⁣(0,τ))3(θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ))8j(\tau) = \frac{32 {\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3}}{{\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}}
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λ(τ)=θ24 ⁣(0,τ)θ34 ⁣(0,τ)\lambda(\tau) = \frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}
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λ(τ)λ(τ)−1=−θ24 ⁣(0,τ)θ44 ⁣(0,τ)\frac{\lambda(\tau)}{\lambda(\tau) - 1} = -\frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{4}^{4}\!\left(0, \tau\right)}
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1−λ(τ)=θ44 ⁣(0,τ)θ34 ⁣(0,τ)1 - \lambda(\tau) = \frac{\theta_{4}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}
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E4 ⁣(τ)=12(θ28 ⁣(0,τ)+θ38 ⁣(0,τ)+θ48 ⁣(0,τ))E_{4}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)
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E6 ⁣(τ)=12(θ312 ⁣(0,τ)+θ412 ⁣(0,τ)−3θ28 ⁣(0,τ)(θ34 ⁣(0,τ)+θ44 ⁣(0,τ)))E_{6}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{3}^{12}\!\left(0, \tau\right) + \theta_{4}^{12}\!\left(0, \tau\right) - 3 \theta_{2}^{8}\!\left(0, \tau\right) \left(\theta_{3}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)\right)\right)

Weierstrass elliptic functions

af0dfc
℘ ⁣(z,τ)=(πθ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(z,τ)θ1 ⁣(z,τ))2−π23(θ24 ⁣(0,τ)+θ34 ⁣(0,τ))\wp\!\left(z, \tau\right) = {\left(\pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left(\theta_{2}^{4}\!\left(0, \tau\right) + \theta_{3}^{4}\!\left(0, \tau\right)\right)
0207dc
ζ ⁣(z,τ)=−z3θ1′′′ ⁣(0,τ)θ1′ ⁣(0,τ)+θ1′ ⁣(z,τ)θ1 ⁣(z,τ)\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} + \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}
b96c9d
σ ⁣(z,τ)=exp⁡ ⁣(−z26θ1′′′ ⁣(0,τ)θ1′ ⁣(0,τ))θ1 ⁣(z,τ)θ1′ ⁣(0,τ)\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)}\right) \frac{\theta_{1}\!\left(z , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)}

Representation by other functions

a9c825
θ2 ⁣(0,τ)=2η2 ⁣(2τ)η(τ)\theta_{2}\!\left(0 , \tau\right) = \frac{2 \eta^{2}\!\left(2 \tau\right)}{\eta(\tau)}
85b2ff
θ3 ⁣(0,τ)=η2 ⁣(12(τ+1))η ⁣(τ+1)\theta_{3}\!\left(0 , \tau\right) = \frac{\eta^{2}\!\left(\frac{1}{2} \left(\tau + 1\right)\right)}{\eta\!\left(\tau + 1\right)}
9448f2
θ4 ⁣(0,τ)=η2 ⁣(12τ)η(τ)\theta_{4}\!\left(0 , \tau\right) = \frac{\eta^{2}\!\left(\frac{1}{2} \tau\right)}{\eta(\tau)}
557b19
θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)=2η3 ⁣(τ)\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) = 2 \eta^{3}\!\left(\tau\right)

Approximations

Main topic: Approximations of Jacobi theta functions
24a793
∣θ3(r) ⁣(z,τ)(2πi)r−(0r+∑n=1N−1nrqn2(w2n+(−1)rw2n))∣≤{2QN2W2NNr1−α,α<1∞,otherwise   where q=eπiτ,  w=eπiz,  Q=∣q∣,  W=max⁡ ⁣(∣w∣,1∣w∣),  α=Q2N+1W2er/N\left|\frac{\theta^{(r)}_{3}\!\left(z , \tau\right)}{{\left(2 \pi i\right)}^{r}} - \left({0}^{r} + \sum_{n=1}^{N - 1} {n}^{r} {q}^{{n}^{2}} \left({w}^{2 n} + \frac{{\left(-1\right)}^{r}}{{w}^{2 n}}\right)\right)\right| \le \begin{cases} \frac{2 {Q}^{{N}^{2}} {W}^{2 N} {N}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} {e}^{r / N}

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