Jacobi Elliptic Functions - Definition As Inverses of Elliptic Integrals

Definition As Inverses of Elliptic Integrals

The above definition, in terms of the unique meromorphic functions satisfying certain properties, is quite abstract. There is a simpler, but completely equivalent definition, giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind. Let

Then the elliptic function sn u is given by

and cn u is given by

and

Here, the angle is called the amplitude. On occasion, dn u = Δ(u) is called the delta amplitude. In the above, the value m is a free parameter, usually taken to be real, 0 ≤ m ≤ 1, and so the elliptic functions can be thought of as being given by two variables, the amplitude and the parameter m.

The remaining nine elliptic functions are easily built from the above three, and are given in a section below.

Note that when, that u then equals the quarter period K.