Abel elliptic functions

Graphical illustration of an elliptic function where its values are indicated by colours. These are periodically repeated in the two directions of the complex plane.

Abel elliptic functions are mathematical functions of one complex variable and with two periods. They were first established by Niels Henrik Abel and are a generalization of trigonometic functions. Since they are based on elliptic integrals, they were the first examples of elliptic functions. Similar functions were shortly thereafter defined by Carl Gustav Jacobi. In spite of the Abel functions having several theoretical advantages, the Jacobi elliptic functions have become the standard. This can have to do with the fact that Abel died only two years after he presented them while Jacobi could continue with his exploration of them throughout his lifetime. Both the elliptic functions of Abel and of Jacobi can be derived from a more general formulation which was later given by Karl Weierstrass based on their double periodicity.

Litterature

• C. Houzel, The Work of Niels Henrik Abel, in O.A. Laudal and R. Piene, The Legacy of Niels Henrik Abel - The Abel Bicentennial, Oslo 2002, Springer Verlag, Berlin (2004). ISBN 3-540-43826-2.