Stumpff function
In celestial mechanics, the Stumpff functions were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation.
They are defined by the alternating series:
Like the sine, cosine, and exponential functions, Stumpf functions are well-behaved entire functions : Their series converge absolutely for any finite argument
Stumpf functions are useful for working with surface launch trajectories, and boosts from closed orbits to escape trajectories, since formulas for spacecraft trajectories using them smoothly meld from conventional closed orbits
Relations to circular and hyperbolic trigononometric functions
By comparing the Taylor series expansion of the trigonometric functions sin and cos with and a relationship can be found. ForSimilarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find for
Circular orbits and elliptical orbits use sine and cosine relations, and hyperbolic orbits use the sinh and cosh relations. Parabolic orbits formulas are a special in-between case.
Recursion
For higher-order Stumpff functions needed for both ordinary trajectories and for perturbation theory, one can use the recurrence relation:or when
Using this recursion, the two further Stumpf functions needed for the universal variable formulation are, for
and for