The Weierstrass Quasi-periodic Functions

Abstract

In the first chapter, we used several times the fact that the derivative of an elliptic function is also an elliptic function with the same periods. However, the opposite statement is not correct; the indefinite integral of an elliptic function is not necessarily an elliptic function. This class of non-elliptic functions typically possess other interesting quasi-periodicity properties. In this chapter we study two such quasi-periodic functions, the zeta and sigma functions, which are derived from the Weierstrass elliptic function. Then, we study their basic properties and express some very useful theorems. Emphasis is given to the theorems that allow the expression of any elliptic function in terms of the aforementioned quasi-periodic functions.