Analyticity of the elliptic functions on the hyperbolic complex plane: exact and duality symmetries for non-linear pendulum

By revealing the exact and duality symmetries for the dynamics of the nonlinear pendulum along new imaginary time directions on the hypercomplex plane, the corresponding analytical continuation and analyticity for the elliptic functions are considered. The known double periodicity of those functions will be extended to a fourfold periodicity on such a plane; this periodicity defines a four dimensional lattice on the hypercomplex plane as solution space, and then it is identified with a four-dimensional torus; the lengths for the S 1 -factors are defined by the elliptic integral periods of the pendulum.