The Jacobi elliptic functions (sn, cn and dn) generalize trigonometric functions to nonlinear systems. They arise naturally from the inversion of elliptic integrals of the first kind and exhibit double periodicity in the complex plane, a central property that sets them apart from elementary functions.
This course builds these functions from the ground up, starting from the geometry of the ellipse: definition via ellipse parameterization, fundamental identities, limiting cases, series expansions, and direct application to nonlinear mechanics. Designed to be accessible from the end of first-year undergraduate level.
References: Complex periodicity (Wikipedia), Jacobi functions (MathWorld), Elliptic integrals (Wikipedia).