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 ) ! − − ⋯ ⋯ = ∑ ∑ n = 0 ∞ ∞ ( − − 1 ) n x n ( k + 2 n ) ! for k = 0 , 1 , 2 , 3 , … … The series above converges absolutely for all real x.