Representation through more general functions

The cosecant function can be represented using more general mathematical functions. As the ratio of one divided by the sine function that is a particular case of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the cosecant 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 cosecant function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the cosecant function when their second parameter is equal to or :