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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.
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