Definitions of Jacobi functions

The Jacobi elliptic amplitude and the twelve Jacobi functions , , , , , , , , , , , and are defined by the following formulas:

It is apparent that the amplitude function is the inverse function to elliptic integral , and the functions , , and are the basic Jacobi functions that are built as the cosine, sine, and derivative of the amplitude function . The other nine Jacobi functions are the ratios of these three basic Jacobi functions or their reciprocal functions.