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Weierstrass elliptic functions

Table of contents: Definitions - Illustrations - Complex lattices - Series and product representations - Derivatives - Theta function representations - 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+∑m2+n2≠01(z+m+nτ)2−1(m+nτ)2\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{{m}^{2} + {n}^{2} \ne 0} \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+∑m2+n2≠01z−m−nτ+1m+nτ+z(m+nτ)2\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{{m}^{2} + {n}^{2} \ne 0} \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∏m2+n2≠0(1−zm+nτ)exp⁡ ⁣(zm+nτ+z22(m+nτ)2)\sigma\!\left(z, \tau\right) = z \prod_{{m}^{2} + {n}^{2} \ne 0} \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)

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((θ2 ⁣(0,τ))4+(θ3 ⁣(0,τ))4)\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({\left(\theta_2\!\left(0, \tau\right)\right)}^{4} + {\left(\theta_3\!\left(0, \tau\right)\right)}^{4}\right)
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ζ ⁣(z,τ)=−z3[d3dz3θ1 ⁣(z,τ)]z=0[ddz θ1 ⁣(z,τ)]z=0+ddz θ1 ⁣(z,τ)θ1 ⁣(z,τ)\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}} + \frac{\frac{d}{d z}\, \theta_1\!\left(z, \tau\right)}{\theta_1\!\left(z, \tau\right)}
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σ ⁣(z,τ)=exp⁡ ⁣(−z26[d3dz3θ1 ⁣(z,τ)]z=0[ddz θ1 ⁣(z,τ)]z=0)θ1 ⁣(z,τ)[ddz θ1 ⁣(z,τ)]z=0\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}

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,τ),z,C)=Λ(1,τ)\operatorname{Poles}\!\left(\wp\!\left(z, \tau\right), z, \mathbb{C}\right) = \Lambda_{(1, \tau)}
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Poles⁡ ⁣(ζ ⁣(z,τ),z,C)=Λ(1,τ)\operatorname{Poles}\!\left(\zeta\!\left(z, \tau\right), z, \mathbb{C}\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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HolomorphicDomain⁡ ⁣(℘ ⁣(z,τ),z,C)=C∖Λ(1,τ)\operatorname{HolomorphicDomain}\!\left(\wp\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C} \setminus \Lambda_{(1, \tau)}
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HolomorphicDomain⁡ ⁣(ζ ⁣(z,τ),z,C)=C∖Λ(1,τ)\operatorname{HolomorphicDomain}\!\left(\zeta\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C} \setminus \Lambda_{(1, \tau)}
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HolomorphicDomain⁡ ⁣(σ ⁣(z,τ),z,C)=C\operatorname{HolomorphicDomain}\!\left(\sigma\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C}

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

2019-06-18 07:49:59.356594 UTC