# 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] They are defined by the formula:

${\displaystyle c_{k}(x)={\frac {1}{k!}}-{\frac {x}{(k+2)!}}+{\frac {x^{2}}{(k+4)!}}-\cdots =\sum _{i=0}^{\infty }{\frac {(-1)^{i}x^{i}}{(k+2i)!}}}$

for ${\displaystyle k=0,1,2,3,\ldots }$ The series above converges absolutely for all real x.

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

${\displaystyle c_{0}(x)=\cos {\sqrt {x}},{\text{ for }}x>0}$
${\displaystyle c_{1}(x)={\frac {\sin {\sqrt {x}}}{\sqrt {x}}},{\text{ for }}x>0}$

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

${\displaystyle c_{0}(x)=\cosh {\sqrt {-x}},{\text{ for }}x<0}$
${\displaystyle c_{1}(x)={\frac {\sinh {\sqrt {-x}}}{\sqrt {-x}}},{\text{ for }}x<0}$

The Stumpff functions satisfy the recursive relations:

${\displaystyle xc_{k+2}(x)={\frac {1}{k!}}-c_{k}(x),{\text{ for }}k=0,1,2,\ldots \,.}$

## References

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