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In celestial mechanics, the Stumpff functions ck(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)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83dd5cce22cd9662ef4cb177a0d17a5c8f1d0631)
for
![{\displaystyle k=0,1,2,3,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0682dbc430868c80201d88de10574907690a1e3)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18bd9b0bfe3b76fdfa6c38e35732181d3475d9df)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0a7280ecfa1fad43790352dcd894f1106baecc)
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 \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b47e7f5105c830c9db702f97a80cf085e5e55bf1)
The Stumpff functions can be expressed in terms of the Mittag-Leffler function:
![{\displaystyle c_{k}(x)=E_{2,k+1}(-x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a028c1bcb0f1d15ea6de340bfd11e4db6d725c2)
References[edit]
- ^ Danby, J.M.A. (1988), Fundamentals of Celestial Mechanics, Willman–Bell, ISBN 9780023271403
- ^ Karl Stumpff (1956), Himmelsmechanik, Deutscher Verlag der Wissenschaften
- ^ Eduard Stiefel, Gerhard Scheifele (1971), Linear and Regular Celestial Mechanics, Springer-Verlag, ISBN 978-0-38705119-2