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In celestial mechanics, the Stumpff functions ck, developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation. They are defined by the formula:
for k = 0 , 1 , 2 , 3 , … {\displaystyle k=0,1,2,3,\ldots } The series above converges absolutely for all real x.
By comparing the Taylor series expansion of the trigonometric functions sin and cos with c0 and c1, a relationship can be found:
Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:
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