An analysis of the large amplitude simple pendulum using Fourier series

The motion of a pendulum is derived using Fourier series and perturbation analysis at levels appropriate for undergraduate physics students. Instead of using the elliptic integral of the first kind, higher order terms of the Taylor-expanded differential equation are considered, leading to increasingly accurate corrections to the period in terms of a single expansion parameter. The relation between the expansion parameter and the initial conditions is not fixed, allowing many solutions to the motion in terms of the expansion parameter but a unique solution in terms of the initial conditions.