Stumpff function

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In celestial mechanics, the Stumpff functions ck(x), developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation.[1] They are defined by the formula:

for The series above converges absolutely for all real x.

By comparing the Taylor series expansion of the trigonometric functions sin and cos with c0(x) and c1(x), a relationship can be found:

Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:

The Stumpff functions satisfy the recursive relations:


  1. ^ Danby, J.M.A (1988), Fundamentals of Celestial Mechanics, Willman–Bell