Representation through more general functions
The secant function can be represented using more general mathematical functions. As the ratio of one and the cosine function that is a particular case of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the secant function can also be represented as ratios of one and those special functions. Here are some examples:
But these representations are not very useful because they include complicated special functions in the denominators.
It is more useful to write the secant function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the secant function when their second parameter is equal to or :