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][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:

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