In celestial mechanics, the Stumpff functions c k ( x ) , {\displaystyle \ c_{k}(x)\ ,} were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation. They are defined by the alternating series:
c k ( x ) ≡ 1 k ! − x ( k + 2 ) ! + x 2 ( k + 4 ) ! − ⋯ = ∑ n = 0 ∞ ( − 1 ) n x n ( k + 2 n ) ! {\displaystyle \ c_{k}(x)~\equiv ~{\frac {1}{\ k!\ }}-{\frac {x}{~\left(k+2\right)!\ }}+{\frac {\ x^{2}}{~\left(k+4\right)!\ }}-\cdots ~=~\sum _{n=0}^{\infty }\ {\frac {\ (-1)^{n}\ x^{n}\ }{\ \left(k+2n\right)!\ }}~~} for k = 0 , 1 , 2 , 3 , … . {\displaystyle ~~k=0,1,2,3,\ \ldots ~~.}
Like the sine, cosine, and exponential functions, Stumpff functions are well-behaved entire functions : Their series converge absolutely for any finite argument x . {\displaystyle \ x~.}
Stumpff functions are useful for working with surface launch trajectories, and boosts from closed orbits to escape trajectories, since formulas for spacecraft trajectories using them smoothly meld from conventional closed orbits (circles and ellipses, eccentricity e : 0 ≤ e < 1 ) to open orbits (parabolas and hyperbolas, ( e ≥ 1 ), with no singularities and no imaginary numbers arising in the expressions as the launch vehicle gains speed to escape velocity and beyond. (The same advantage occurs in reverse, as a spacecraft decelerates from an arrival trajectory to go into a closed orbit around its destination, or descends to a planet's surface from a stable orbit.)
Relations to circular and hyperbolic trigononometric functions By comparing the Taylor series expansion of the trigonometric functions sin and cos with c 0 ( x ) {\displaystyle \ c_{0}(x)\ } and c 1 ( x ) , {\displaystyle \ c_{1}(x)\ ,} a relationship can be found. For x > 0 : {\displaystyle ~x>0\ :}
c 0 ( x ) = cos x , c 1 ( x ) = sin x x . {\displaystyle {\begin{aligned}c_{0}(x)~&=~~\cos {\sqrt {x\ }}\ ,\\[1ex]c_{1}(x)~&=~{\frac {\ \sin {\sqrt {x\ }}\ }{\sqrt {x\ }}}~.\end{aligned}}}
Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find for x < 0 : {\displaystyle ~x<0\ :}
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