# $\wp$-function

Let $L$ be a lattice on $\mathbbmss{C}$. 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 $\mathbbmss{C}\setminus L$, 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

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

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

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

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