# Weierstrass elliptic functions

## Definitions

Symbol: WeierstrassP $\wp\!\left(z, \tau\right)$ Weierstrass elliptic function
Symbol: WeierstrassZeta $\zeta\!\left(z, \tau\right)$ Weierstrass zeta function
Symbol: WeierstrassSigma $\sigma\!\left(z, \tau\right)$ Weierstrass sigma function

## Illustrations

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

## Complex lattices

Symbol: Lattice $\Lambda_{(a, b)}$ Complex lattice with periods a, b
$\Lambda_{(a, b)} = \left\{ a m + b n : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$

## Series and product representations

$\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{{m}^{2} + {n}^{2} \ne 0} \frac{1}{{\left(z + m + n \tau\right)}^{2}} - \frac{1}{{\left(m + n \tau\right)}^{2}}$
$\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{{m}^{2} + {n}^{2} \ne 0} \frac{1}{z - m - n \tau} + \frac{1}{m + n \tau} + \frac{z}{{\left(m + n \tau\right)}^{2}}$
$\sigma\!\left(z, \tau\right) = z \prod_{{m}^{2} + {n}^{2} \ne 0} \left(1 - \frac{z}{m + n \tau}\right) \exp\!\left(\frac{z}{m + n \tau} + \frac{{z}^{2}}{2 {\left(m + n \tau\right)}^{2}}\right)$

## Derivatives

$\frac{d}{d z}\, \zeta\!\left(z, \tau\right) = -\wp\!\left(z, \tau\right)$
$\frac{d}{d z}\, \sigma\!\left(z, \tau\right) = \zeta\!\left(z, \tau\right) \sigma\!\left(z, \tau\right)$

## Theta function representations

$\wp\!\left(z, \tau\right) = {\left(\pi \theta_2\!\left(0, \tau\right) \theta_3\!\left(0, \tau\right) \frac{\theta_4\!\left(z, \tau\right)}{\theta_1\!\left(z, \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left({\left(\theta_2\!\left(0, \tau\right)\right)}^{4} + {\left(\theta_3\!\left(0, \tau\right)\right)}^{4}\right)$
$\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}} + \frac{\frac{d}{d z}\, \theta_1\!\left(z, \tau\right)}{\theta_1\!\left(z, \tau\right)}$
$\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}$

## Symmetries

$\wp\!\left(-z, \tau\right) = \wp\!\left(z, \tau\right)$
$\zeta\!\left(-z, \tau\right) = -\zeta\!\left(z, \tau\right)$
$\sigma\!\left(-z, \tau\right) = -\sigma\!\left(z, \tau\right)$

## Periodicity

$\wp\!\left(z + m + n \tau, \tau\right) = \wp\!\left(z, \tau\right)$
$\zeta\!\left(z + 1, \tau\right) = \zeta\!\left(z, \tau\right) + \zeta\!\left(\frac{1}{2}, \tau\right)$
$\zeta\!\left(z + \tau, \tau\right) = \zeta\!\left(z, \tau\right) + \zeta\!\left(\frac{\tau}{2}, \tau\right)$
$\sigma\!\left(z + 1, \tau\right) = -\exp\!\left(2 \left(z + \frac{1}{2}\right) \zeta\!\left(\frac{1}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)$
$\sigma\!\left(z + \tau, \tau\right) = -\exp\!\left(2 \left(z + \frac{\tau}{2}\right) \zeta\!\left(\frac{\tau}{2}, \tau\right)\right) \sigma\!\left(z, \tau\right)$

## Analytic properties

$\operatorname{Poles}\!\left(\wp\!\left(z, \tau\right), z, \mathbb{C}\right) = \Lambda_{(1, \tau)}$
$\operatorname{Poles}\!\left(\zeta\!\left(z, \tau\right), z, \mathbb{C}\right) = \Lambda_{(1, \tau)}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \sigma\!\left(z, \tau\right) = \Lambda_{(1, \tau)}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \wp\!\left(z, i\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) i : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\operatorname{HolomorphicDomain}\!\left(\wp\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C} \setminus \Lambda_{(1, \tau)}$
$\operatorname{HolomorphicDomain}\!\left(\zeta\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C} \setminus \Lambda_{(1, \tau)}$
$\operatorname{HolomorphicDomain}\!\left(\sigma\!\left(z, \tau\right), z, \mathbb{C}\right) = \mathbb{C}$

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