Stumpff function

In <a href="/facts/Celestial_mechanics/FA3uR78J">celestial mechanics</a>, the Stumpff functions 
 
 
 
  
 
 c
 
 k
 
 
 (
 x
 )
  
 ,
 
 
 {\displaystyle \ c_{k}(x)\ ,}
 
 were developed by <a href="/facts/Karl_Stumpff/GoDtn0Ev">Karl Stumpff</a> for analyzing trajectories and <a href="/facts/Orbit/nD2TPySG">orbits</a> using the <a href="/facts/Universal_variable_formulation/nXGMrHA8">universal variable formulation</a>.
They are defined by the alternating <a href="/facts/Taylor_series/M0aTuftH">series</a>:

c
          
            k
          
        
        (
        x
        )
         
        ≡
         
        
          
            1
            
               
              k
              !
               
            
          
        
        −
        
          
            x
            
               
              
                (
                
                  k
                  +
                  2
                
                )
              
              !
               
            
          
        
        +
        
          
            
               
              
                x
                
                  2
                
              
            
            
               
              
                (
                
                  k
                  +
                  4
                
                )
              
              !
               
            
          
        
        −
        ⋯
         
        =
         
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
         
        
          
            
               
              (
              −
              1
              
                )
                
                  n
                
              
               
              
                x
                
                  n
                
              
               
            
            
               
              
                (
                
                  k
                  +
                  2
                  n
                
                )
              
              !
               
            
          
        
         
         
      
    
    {\displaystyle \ c_{k}(x)~\equiv ~{\frac {1}{\ k!\ }}-{\frac {x}{~\left(k+2\right)!\ }}+{\frac {\ x^{2}}{~\left(k+4\right)!\ }}-\cdots ~=~\sum _{n=0}^{\infty }\ {\frac {\ (-1)^{n}\ x^{n}\ }{\ \left(k+2n\right)!\ }}~~}
  
 for 
  
    
      
         
         
        k
        =
        0
        ,
        1
        ,
        2
        ,
        3
        ,
         
        …
         
         
        .
      
    
    {\displaystyle ~~k=0,1,2,3,\ \ldots ~~.}

Like the <a href="/facts/Sine/gQ6LXK8z">sine</a>, <a href="/facts/Cosine/czej7ur0">cosine</a>, and <a href="/facts/Exponential_function/ewlYxvR2">exponential</a> functions, Stumpf functions are well-behaved <a href="/facts/Entire_function/XNluKzu7">entire functions</a> : Their <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> <a href="/facts/Absolute_convergence/BXz9kBJ5">converge absolutely</a> for any <a href="/facts/Glossary_of_mathematical_jargon/fTRoAtn2">finite</a> <a href="/facts/Function_argument/LcuLbbSq">argument</a> 
 
 
 
  
 x
  
 .
 
 
 {\displaystyle \ x~.}

Stumpf functions are useful for working with surface <a href="/facts/Rocket_launch/b6j4keIl">launch</a> trajectories, and boosts from closed orbits to <a href="/facts/Escape_orbit/vDNtF83E">escape trajectories</a>, since <a href="/facts/Universal_variable_formulation/nXGMrHA8">formulas for spacecraft trajectories using them</a> smoothly meld from conventional closed orbits (circles and ellipses, <a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">eccentricity</a> e : 0 ≤ e < 1 ) to open orbits (parabolas and hyperbolas, ( e ≥ 1 ), with no singularities and no <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary numbers</a> arising in the expressions as the launch vehicle gains speed to escape velocity and beyond. (The same advantage occurs in reverse, as a spacecraft decelerates from an arrival trajectory to go into a closed orbit around its destination, or descends to a planet's surface from a stable orbit.)

Stumpff function open-in-new

Stumpff function