Elliptic function
From Conservapedia
A function is an Elliptic function if and only if it is meromorphic and doubly periodic. Elliptic functions are the inverse functions of elliptic integrals. Elliptic functions arise in the study of Schwarz-Christoffel transformations of functions mapping the upper-half plane conformally into the interior of a rectangle. One elliptic function, the Weierstrass p-function, with periods ω1 and ω2 is given by
![\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+
\sum_{m^2+n^2 \ne 0}
\left [
\frac{1}{(z-m\omega_1-n\omega_2)^2}-
\frac{1}{\left(m\omega_1+n\omega_2\right)^2}
\right ].](images/math/8/0/f/80feda3fec1dec029395edcbeb439b25.png)
The Weierstrass function and its derivative together generate the field of elliptic functions with given periods.
The derivative of an elliptic function is also an elliptic function with the same periods. The set of all elliptic functions with the same fundamental periods form a field. The number of zeros (counted with multiplicity) in any fundamental parallelogram is the order of the elliptic function. The number of poles in any fundamental parallelogram is finite, and unless the elliptic function is constant, any fundamental parallelogram has at least one pole, which is a consequence of Liouville's theorem.
