The Schwarz-Christoffel transformation and elliptic functions

An important result from complex analysis, the Riemann mapping theorem, states that there exists a conformal bijective mapping f, from A to B between any two simply connected open sets A and B, both not equal to the whole complex-plane C. In the case where the upper half-plane is conformally mapped onto an open set which is the inside of a simple polygon, the mapping has the form of a Schwarz-Christoffel transformation (SCT). The SCT will be discussed in detail, and will be used to define Jacobi elliptic functions. Jacobi elliptic functions form a special set of elliptic functions in general. Elliptic functions are doubly periodic meromorphic functions. Basic properties of elliptic functions will be discussed.