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Weierstrass elliptic functions

Table of contents: Definitions - Illustrations - Complex lattices - Series and product representations - Derivatives - Theta function representations - Inverse functions - Symmetries - Periodicity - Analytic properties

Definitions

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Symbol: WeierstrassP — ℘ ⁣(z,τ)\wp\!\left(z, \tau\right) — Weierstrass elliptic function
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Symbol: WeierstrassZeta — ζ ⁣(z,τ)\zeta\!\left(z, \tau\right) — Weierstrass zeta function
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Symbol: WeierstrassSigma — σ ⁣(z,τ)\sigma\!\left(z, \tau\right) — Weierstrass sigma function

Illustrations

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Image: X-ray of ℘ ⁣(z,i)\wp\!\left(z, i\right) on [−1.5,1.5]+[−1.5,1.5]i\left[-1.5, 1.5\right] + \left[-1.5, 1.5\right] i with lattice cell highlighted
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Image: X-ray of ℘ ⁣(z,eπi/3)\wp\!\left(z, {e}^{\pi i / 3}\right) on z∈[−1.5,1.5]+[−1.5,1.5]iz \in \left[-1.5, 1.5\right] + \left[-1.5, 1.5\right] i with lattice cell highlighted
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Image: X-ray of ℘ ⁣(z,−0.8+0.7i)\wp\!\left(z, -0.8 + 0.7 i\right) on z∈[−1.5,1.5]+[−1.5,1.5]iz \in \left[-1.5, 1.5\right] + \left[-1.5, 1.5\right] i with lattice cell highlighted

Complex lattices

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Symbol: Lattice — Λ(a,b)\Lambda_{(a, b)} — Complex lattice with periods a, b
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Λ(a,b)={am+bn:m∈Z  and⁡  n∈Z}\Lambda_{(a, b)} = \left\{ a m + b n : m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

Series and product representations

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℘ ⁣(z,τ)=1z2+∑(m,n)∈Z2∖{(0,0)}1(z+m+nτ)2−1(m+nτ)2\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(z + m + n \tau\right)}^{2}} - \frac{1}{{\left(m + n \tau\right)}^{2}}
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ζ ⁣(z,τ)=1z+∑(m,n)∈Z2∖{(0,0)}1z−m−nτ+1m+nτ+z(m+nτ)2\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{z - m - n \tau} + \frac{1}{m + n \tau} + \frac{z}{{\left(m + n \tau\right)}^{2}}
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σ ⁣(z,τ)=z∏(m,n)∈Z2∖{(0,0)}(1−zm+nτ)exp⁡ ⁣(zm+nτ+z22(m+nτ)2)\sigma\!\left(z, \tau\right) = z \prod_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \left(1 - \frac{z}{m + n \tau}\right) \exp\!\left(\frac{z}{m + n \tau} + \frac{{z}^{2}}{2 {\left(m + n \tau\right)}^{2}}\right)
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℘ ⁣(z,τ)=1z2+∑k=1∞(2k+1)G2k+2 ⁣(τ)z2k\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}
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ζ ⁣(z,τ)=1z−∑k=1∞G2k+2 ⁣(τ)z2k+1\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}

Derivatives

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ddz ζ ⁣(z,τ)=−℘ ⁣(z,τ)\frac{d}{d z}\, \zeta\!\left(z, \tau\right) = -\wp\!\left(z, \tau\right)
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ddz σ ⁣(z,τ)=ζ ⁣(z,τ)σ ⁣(z,τ)\frac{d}{d z}\, \sigma\!\left(z, \tau\right) = \zeta\!\left(z, \tau\right) \sigma\!\left(z, \tau\right)

Theta function representations

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℘ ⁣(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)
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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)}
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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)}

Inverse functions

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℘ ⁣(f(z),τ)=z   where f(z)=RF ⁣(z−e1 ⁣(τ),z−e2 ⁣(τ),z−e3 ⁣(τ))\wp\!\left(f(z), \tau\right) = z\; \text{ where } f(z) = R_F\!\left(z - e_{1}\!\left(\tau\right), z - e_{2}\!\left(\tau\right), z - e_{3}\!\left(\tau\right)\right)

Symmetries

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℘ ⁣(−z,τ)=℘ ⁣(z,τ)\wp\!\left(-z, \tau\right) = \wp\!\left(z, \tau\right)
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ζ ⁣(−z,τ)=−ζ ⁣(z,τ)\zeta\!\left(-z, \tau\right) = -\zeta\!\left(z, \tau\right)
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σ ⁣(−z,τ)=−σ ⁣(z,τ)\sigma\!\left(-z, \tau\right) = -\sigma\!\left(z, \tau\right)

Periodicity

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℘ ⁣(z+m+nτ,τ)=℘ ⁣(z,τ)\wp\!\left(z + m + n \tau, \tau\right) = \wp\!\left(z, \tau\right)
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ζ ⁣(z+1,τ)=ζ ⁣(z,τ)+ζ ⁣(12,τ)\zeta\!\left(z + 1, \tau\right) = \zeta\!\left(z, \tau\right) + \zeta\!\left(\frac{1}{2}, \tau\right)
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ζ ⁣(z+τ,τ)=ζ ⁣(z,τ)+ζ ⁣(τ2,τ)\zeta\!\left(z + \tau, \tau\right) = \zeta\!\left(z, \tau\right) + \zeta\!\left(\frac{\tau}{2}, \tau\right)
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σ ⁣(z+1,τ)=−exp⁡ ⁣(2(z+12)ζ ⁣(12,τ))σ ⁣(z,τ)\sigma\!\left(z + 1, \tau\right) = -\exp\!\left(2 \left(z + \frac{1}{2}\right) \zeta\!\left(\frac{1}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)
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σ ⁣(z+τ,τ)=−exp⁡ ⁣(2(z+τ2)ζ ⁣(τ2,τ))σ ⁣(z,τ)\sigma\!\left(z + \tau, \tau\right) = -\exp\!\left(2 \left(z + \frac{\tau}{2}\right) \zeta\!\left(\frac{\tau}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)

Analytic properties

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poles⁡ z∈C℘ ⁣(z,τ)=Λ(1,τ)\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \wp\!\left(z, \tau\right) = \Lambda_{(1, \tau)}
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poles⁡ z∈Cζ ⁣(z,τ)=Λ(1,τ)\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \zeta\!\left(z, \tau\right) = \Lambda_{(1, \tau)}
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zeros⁡ z∈Cσ ⁣(z,τ)=Λ(1,τ)\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \sigma\!\left(z, \tau\right) = \Lambda_{(1, \tau)}
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zeros⁡ z∈C℘ ⁣(z,i)={(m+12)+(n+12)i:m∈Z  and⁡  n∈Z}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \wp\!\left(z, i\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) i : m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}
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℘ ⁣(z,τ) is holomorphic on z∈C∖Λ(1,τ)\wp\!\left(z, \tau\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \Lambda_{(1, \tau)}
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ζ ⁣(z,τ) is holomorphic on z∈C∖Λ(1,τ)\zeta\!\left(z, \tau\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \Lambda_{(1, \tau)}
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σ ⁣(z,τ) is holomorphic on z∈C\sigma\!\left(z, \tau\right) \text{ is holomorphic on } z \in \mathbb{C}

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