Elliptic function

One of the most important elliptic functions is the Weierstrass ${\displaystyle \wp }$-function. For a given period lattice ${\displaystyle \Lambda }$ it is defined by


It is constructed in such a way that it has a pole of order two at every lattice point. The term ${\displaystyle -{\frac {1}{\lambda ^{2}}}}$ is there to make the series convergent.


${\displaystyle \wp }$ is an even elliptic function; that is, ${\displaystyle \wp (-z)=\wp (z)}$.^[6]


Its derivative


is an odd function, i.e. ${\displaystyle \wp '(-z)=-\wp '(z).}$^[6]


One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice ${\displaystyle \Lambda }$ can be expressed as a rational function in terms of ${\displaystyle \wp }$ and ${\displaystyle \wp '}$.^[7]


The ${\displaystyle \wp }$-function satisfies the differential equation


where ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are constants that depend on ${\displaystyle \Lambda }$. More precisely, ${\displaystyle g_{2}(\omega _{1},\omega _{2})=60G_{4}(\omega _{1},\omega _{2})}$ and ${\displaystyle g_{3}(\omega _{1},\omega _{2})=140G_{6}(\omega _{1},\omega _{2})}$, where ${\displaystyle G_{4}}$ and ${\displaystyle G_{6}}$ are so called Eisenstein series.^[8]


In algebraic language, the field of elliptic functions is isomorphic to the field


where the isomorphism maps ${\displaystyle \wp }$ to ${\displaystyle X}$ and ${\displaystyle \wp '}$ to ${\displaystyle Y}$.