Notation
The elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing. Elliptic functions are functions of two variables. The first variable might be given in terms of the amplitude φ, or more commonly, in terms of u given below. The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where k2 = m, or in terms of the modular angle α, where m = sin2 α. A more extensive review and definition of these alternatives, their complements, and the associated notation schemes are given in the articles on elliptic integrals and quarter period.
Read more about this topic: Jacobi Elliptic Functions