Stumpff function

In celestial mechanics, the Stumpff functions ck(x), developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation.[1][2][3] 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 recurrence relation:

The Stumpff functions can be expressed in terms of the Mittag-Leffler function:

References

  1. Danby, J.M.A. (1988), Fundamentals of Celestial Mechanics, Willman–Bell, ISBN 9780023271403
  2. Karl Stumpff (1956), Himmelsmechanik, Deutscher Verlag der Wissenschaften
  3. Eduard Stiefel, Gerhard Scheifele (1971), Linear and Regular Celestial Mechanics, Springer-Verlag, ISBN 978-0-38705119-2


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