Stumpff function - Alchetron, The Free Social Encyclopedia

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 ) = 1 k ! − x ( k + 2 ) ! + x 2 ( k + 4 ) ! − ⋯ = ∑ i = 0 ∞ ( − 1 ) i x i ( k + 2 i ) !

for k = 0 , 1 , 2 , 3 , … 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:

c 0 ( x ) = cos ⁡ x , for x > 0 c 1 ( x ) = sin ⁡ x x , for x > 0

Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:

c 0 ( x ) = cosh ⁡ − x , for x < 0 c 1 ( x ) = sinh ⁡ − x − x , for x < 0

The Stumpff functions satisfy the recursive relations:

x c k + 2 ( x ) = 1 k ! − c k ( x ) , for k = 0 , 1 , 2 , … .

References

Stumpff function Wikipedia

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