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Elliptic Functions > JacobiCS[z,m] > Introduction to the Jacobi elliptic functions

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.