Stumpff function

In celestial mechanics, the Stumpff functionsc_k(x), developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation. ^{[1] [2] [3]} They are defined by the formula:

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

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:

$${\displaystyle {\begin{aligned}c_{0}(x)&=\cos {\sqrt {x}},\\[1ex]c_{1}(x)&={\frac {\sin {\sqrt {x}}}{\sqrt {x}}},\end{aligned}}\quad {\text{ for }}x>0}$$

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

$${\displaystyle {\begin{aligned}c_{0}(x)&=\cosh {\sqrt {-x}},\\[1ex]c_{1}(x)&={\frac {\sinh {\sqrt {-x}}}{\sqrt {-x}}},\end{aligned}}\quad {\text{ for }}x<0}$$

The Stumpff functions satisfy the recurrence relation:

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

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

$${\displaystyle c_{k}(x)=E_{2,k+1}(-x).}$$

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