Expanding the simple pendulum's rotation solution in action-angle variables

Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions, the transformation to action-angle variables may need to remain in implicit form. This is exactly the case of the simple pendulum, where it is shown that in order to make explicit the transformation to action-angle variables, one needs to resort to nontrivial expansions of special functions and series reversion.