On The Power Series Solution to The Nonlinear Pendulum

The exact solution to the simple pendulum problem has long been known in terms of Jacobi elliptic functions, of which an efficient numerical evaluation is standard in most scientific computing software packages. Alternatively, and as done in V. Fairén, López and L. Conde (Power series approximation to solutions of nonlinear systems of differential equation, Am. J. Phys. 56 (1988) 57–61], the pendulum equation can be analytically solved exactly by the power series solution method. Although recursive formulae for the series coefficients were provided in V. Fairén et al., the series itself—as well as the optimal location about which an expansion should be chosen to maximize the domain of convergence—has not yet been examined, and this is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the power series solution in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. We provide an exact resummation of the pendulum series that accelerates the series’ convergence uniformly from the top to the bottom of the trajectory. We also provide the formulae needed to extend the pendulum series for all time via symmetry. The pendulum problem, in its relative simplicity, provides an explicit demonstration of the effect of singularity structure and initial condition location on convergence properties of power series solutions.