Astronomy:Stumpff function
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: [math]\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)!}} }[/math] for [math]\displaystyle{ k = 0, 1, 2, 3,\ldots }[/math] 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: [math]\displaystyle{ \begin{align} c_0(x) &= \cos {\sqrt x}, \\[1ex] c_1(x) &= \frac{\sin {\sqrt x}}{\sqrt x}, \end{align} \quad \text{ for }x \gt 0 }[/math] Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find: [math]\displaystyle{ \begin{align} c_0(x) &= \cosh {\sqrt {-x}}, \\[1ex] c_1(x) &= \frac{\sinh {\sqrt {-x}}}{\sqrt {-x}}, \end{align} \quad \text{ for }x \lt 0 }[/math]
The Stumpff functions satisfy the recurrence relation: [math]\displaystyle{ x c_{k+2}(x) = \frac{1}{k!} - c_k(x),\text{ for }k = 0, 1, 2, \ldots\,. }[/math]
The Stumpff functions can be expressed in terms of the Mittag-Leffler function:
[math]\displaystyle{ c_{k}(x) = E_{2,k+1}(-x). }[/math]
References
- ↑ 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
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