In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

A cubic of the form ${\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}}$, where ${\displaystyle g_{2},g_{3}\in \mathbb {C} }$ are complex numbers with ${\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}$, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric ${\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}}$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: $${\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).}$$ Because of the periodicity of the sine and cosine ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$ is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of ${\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}$ by means of the doubly periodic ${\displaystyle \wp }$-function and its derivative, namely via ${\displaystyle (x,y)=(\wp (z),\wp '(z))}$. This parameterization has the domain ${\displaystyle \mathbb {C} /\Lambda }$, which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. Consider the integral function $${\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.}$$ It can be simplified by substituting ${\displaystyle y=\sin t}$ and ${\displaystyle s=\arcsin x}$: $${\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.}$$ That means ${\displaystyle a^{-1}(x)=\sin x}$. So the sine function is an inverse function of an integral function.

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $${\displaystyle u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.}$$ Then the extension of ${\displaystyle u^{-1}}$ to the complex plane equals the ${\displaystyle \wp }$-function. This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.

Let ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }$ be two complex numbers that are linearly independent over ${\displaystyle \mathbb {R} }$ and let ${\displaystyle \Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}$ be the period lattice generated by those numbers. Then the ${\displaystyle \wp }$-function is defined as follows:

This series converges locally uniformly absolutely in the complex torus ${\displaystyle \mathbb {C} /\Lambda }$.