# 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

$\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{{\left(z + m + n \tau\right)}^{2}} - \frac{1}{{\left(m + n \tau\right)}^{2}}$
$\zeta\!\left(z, \tau\right) = \frac{1}{z} + \sum_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \frac{1}{z - m - n \tau} + \frac{1}{m + n \tau} + \frac{z}{{\left(m + n \tau\right)}^{2}}$
$\sigma\!\left(z, \tau\right) = z \prod_{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\}} \left(1 - \frac{z}{m + n \tau}\right) \exp\!\left(\frac{z}{m + n \tau} + \frac{{z}^{2}}{2 {\left(m + n \tau\right)}^{2}}\right)$
$\wp\!\left(z, \tau\right) = \frac{1}{{z}^{2}} + \sum_{k=1}^{\infty} \left(2 k + 1\right) G_{2 k + 2}\!\left(\tau\right) {z}^{2 k}$
$\zeta\!\left(z, \tau\right) = \frac{1}{z} - \sum_{k=1}^{\infty} G_{2 k + 2}\!\left(\tau\right) {z}^{2 k + 1}$

## 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

$\wp\!\left(z, \tau\right) = {\left(\pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}\right)}^{2} - \frac{{\pi}^{2}}{3} \left(\theta_{2}^{4}\!\left(0, \tau\right) + \theta_{3}^{4}\!\left(0, \tau\right)\right)$
$\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} + \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}$
$\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)}\right) \frac{\theta_{1}\!\left(z , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)}$

## Inverse functions

$\wp\!\left(f(z), \tau\right) = z\; \text{ where } f(z) = R_F\!\left(z - e_{1}\!\left(\tau\right), z - e_{2}\!\left(\tau\right), z - e_{3}\!\left(\tau\right)\right)$

## Symmetries

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

## 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

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

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

2020-04-08 16:14:44.404316 UTC