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Elementary Functions > Tanh[z] > Introduction to the Hyperbolic Tangent Function

Representation through more general functions

The hyperbolic tangent function can be represented using more general mathematical functions. As the ratio of the hyperbolic sine and cosine functions that are particular cases of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the hyperbolic tangent function can also be represented as ratios of those special functions. But these representations are not very useful. It is more useful to write the hyperbolic tangent function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the hyperbolic tangent function when their second parameter is equal to or :