$\mathrm{\wp }$-function

Let $L$ be a lattice on $ℂ$. The Weierstrass $\mathrm{\wp }$ function associated to $L$ is given by

$$\mathrm{\wp }(z)=\frac{1}{z^2}+\sum _{w\in L\setminus \{0\}}\left(\frac{1}{{(z-w)}^2}-\frac{1}{w^2}\right).$$

The $\mathrm{\wp }$ function is meromorphic and analytic on $ℂ\setminus L$, whereas at each $w\in L$, it has an order $2$ pole. It is also an even function, because $\mathrm{\wp }(z)=\mathrm{\wp }(-z)$.

Its derivative

$${\mathrm{\wp }}^{\prime }(z)=-2\sum _{w\in L}\frac{1}{{(z-w)}^3}$$

is also an odd, meromorphic, and elliptic function, analytic at $ℂ\setminus L$ and having order $3$ poles at each $w\in L$.

The functions $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\prime }$ form together a generator set for the field of elliptic functions associated to the lattice $L$.

Title	$\mathrm{\wp }$-function
Canonical name	wpfunction
Date of creation	2013-03-22 14:51:54
Last modified on	2013-03-22 14:51:54
Owner	drini (3)
Last modified by	drini (3)
Numerical id	10
Author	drini (3)
Entry type	Definition
Classification	msc 33E05
Synonym	$\mathrm{\wp }$
Synonym	Weierstrass $\mathrm{\wp }$ function
Synonym	Weierstrass p-function
Synonym	p-Weierstrass
Synonym	Weierstrass $\mathrm{\wp }$-function
Related topic	EllipticCurve
Related topic	EllipticFunction
Related topic	ExamplesOfEllipticFunctions