Franklin's Notes


Half-period values of the P function

The differential equation for the Weierstrass P function $\wp$ is as follows: for some fixed values of the periods $\omega_1,\omega_2$. The RHS is a cubic in $\wp$, meaning that it can be factored as so that $e_1,e_2,e_3$ are the values of $\wp$ when $\wp'=0$, i.e. at the critical points of the $\wp$ function in a fundamental parallelogram. We are also able to determine explicitly exactly where $\wp$ has its critical values. Because $\wp$ is even, we have that meaning that we have and therefore whenever $\omega$ is a period of $\wp$. If we set $z=\omega/2$ in this equation, we find that and therefore Thus, the $\wp$ function has a critical point at every half-period value where it does not have a pole. In the fundamental parallelogram with one vertex at the origin, these half-period values are $\omega_1/2$, $\omega_2/2$, and $(\omega_1+\omega_2)/2$. Hence, $e_1,e_2$, and $e_3$ are defined as functions of $\omega_1,\omega_2$ as follows: Often, we consider the $\wp$ function with $\omega_1=1$ and $\omega_2=\tau\notin\mathbb R$, so that we have

complex-analysis

elliptic-functions

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