The variable ϕ (also denoted am(u, k)) used in elliptic functions and elliptic integrals is called the amplitude (or Jacobi amplitude). It can be defined by ϕ | = | am(u, k) | = | integral_0^u dn(u', k) d u', where dn(u, k) is a Jacobi elliptic function with elliptic modulus. As is common with Jacobi elliptic functions, the modulus k is often suppressed for conciseness. The Jacobi amplitude is the inverse function of the elliptic integral of the first kind. The amplitude function is implemented in the Wolfram Language as JacobiAmplitude[u, m], where m = k^2 is the parameter.
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