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. They are defined by the formula:
$c_k (x) = \frac - \frac + \frac - \cdots = \sum_^\infty$
for $k = 0, 1, 2, 3,\ldots$ The series above converges absolutely for all real ''x''.

By comparing the Taylor series expansion of the trigonometric functions sin and cos with ''c''₀(''x'') and ''c''₁(''x''), a relationship can be found:
\begin
c_0(x) &= \cos , \\ exc_1(x) &= \frac,
\end \quad \textx > 0
Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:
\begin
c_0(x) &= \cosh , \\ exc_1(x) &= \frac,
\end \quad \textx < 0

The Stumpff functions satisfy the recurrence relation:
$x c_(x) = \frac - c_k(x),\textk = 0, 1, 2, \ldots\,.$

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

$c_(x) = E_(-x).$

References

Orbits

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