Stumpff function

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

$c_k (x) = \frac{1}{k!} - \frac{x}{(k + 2)!} + \frac{x^2}{(k + 4)!} - \cdots = \sum_{i=0}^\infty {\frac{(-1)^i x^i}{(k + 2i)!}}$

for $k = 0, 1, 2, 3,\ldots$ 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:

$c_0(x) = \cos {\sqrt x},\text{ for }x > 0$

$c_1(x) = \frac{\sin {\sqrt x}}{\sqrt x},\text{ for }x > 0$

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

$c_0(x) = \cosh {\sqrt {-x}},\text{ for }x < 0$

$c_1(x) = \frac{\sinh {\sqrt {-x}}}{\sqrt {-x}},\text{ for }x < 0$

The Stumpff functions satisfy the recursive relations:

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

References

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