A complex exponential Keplerian universal solution
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An alternative solution to Battin's universal solution of the two-body problem has been developed using complex exponential functions. This Complex Exponential Keplerian Solution (CEKU) is an exact solution that can be efficiently implemented for numerical computations. Analogous to the classical universal developments by Batting,1 Herrick, Stiefel,2 et al, this formulation eliminates singularities associated with the elliptical and hyperbolic trajectories that arise at zero eccentricity and zero inclination. Also, a single, unified solution form holds for both elliptic and hyperbolic orbits. The parabolic case is a singularity in the CEKU's current form, but the singularity can be eliminated with a power series for near parabolic trajectories. In lieu of using the Stumpff and related universal functions, we utilize the usual exponential functions with complex arguments. As a consequence of the special structure that flows from this approach, we find that this solution is an elegant, unified alternative to the classical universal development. Our developments lead to new forms for the Lagrange-Gibbs F and G functions, a Universal Kepler equation, and also the state transition matrix. We present the formulations and compare/contrast them with the classical developments.