Stumpff function

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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:

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\,.


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