Let $L$ be a lattice on . The Weierstrass $\wp$ function associated to $L$ is given by$$ \wp(z) = \frac{1}{z^2} + \sum_{w\in L\setminus\{0\}} \left(\frac{1}{(z-w)^2} - \frac{1}{w^2} \right).$$

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

Its derivative$$ \wp'(z)=-2\sum_{w\in L} \frac{1}{(z-w)^3}$$ is also an odd, meromorphic, and elliptic function, analytic at and having order $3$ poles at each $w\in L$ .

The functions $\wp$ and $\wp'$ form together a generator set for the field of elliptic functions associated to the lattice $L$ .