Stumpff function
Encyclopedia
In celestial mechanics
, the Stumpff functions ck(x), developed by Karl Stumpff
, are used for analyzing orbit
s using the universal variable formulation
. They are defined by the formula:
for
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:
, for 
, for 
Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:
, for 
, for 
The Stumpff functions satisfy the recursive
relations:
, for 
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, the Stumpff functions ck(x), developed by Karl Stumpff
Karl Stumpff
Karl Johann Nikolaus Stumpff was a German astronomer. The Stumpff functions, used in the universal variable formulation of the two-body problem, are named after him.-External links:*....
, are used for analyzing orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s using the universal variable formulation
Universal variable formulation
In orbital mechanics, the universal variable formulation is a method used to solve the two-body Kepler problem. It is a generalized form of Kepler's Equations, extending them to apply not only to elliptic orbits, but also parabolic and hyperbolic orbits...
. They are defined by the formula:
for
The seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
above converges absolutely
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
for all real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
x.
By comparing the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
expansion of the trigonometric functions sin and cos with c0(x) and c1(x), a relationship can be found:
, for , for Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find:
, for , for The Stumpff functions satisfy the recursive
Recursive definition
In mathematical logic and computer science, a recursive definition is used to define an object in terms of itself ....
relations:
, for