Stumpff function

<h2 id="relations-to-circular-and-hyperbolic-trigononometric-functions">Relations to circular and hyperbolic trigononometric functions</h2>
<p>By comparing the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> expansion of the <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a> sin and cos with 
  
    
      
         
        
          c
          
            0
          
        
        (
        x
        )
         
      
    
    {\displaystyle \ c_{0}(x)\ }
  
 and 
  
    
      
         
        
          c
          
            0
          
        
        (
        x
        )
         
        ,
      
    
    {\displaystyle \ c_{0}(x)\ ,}
  
 a relationship can be found. For 
  
    
      
         
        x
        >
        0
         
        :
      
    
    {\displaystyle ~x>0\ :}

</p>

c
                  
                    0
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                 
                cos
                ⁡
                
                  
                    x
                     
                  
                
                 
                ,
              
            
            
              
                
                  c
                  
                    1
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                
                  
                    
                       
                      sin
                      ⁡
                      
                        
                          x
                           
                        
                      
                       
                    
                    
                      x
                       
                    
                  
                
                 
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}c_{0}(x)~&=~~\cos {\sqrt {x\ }}\ ,\\[1ex]c_{1}(x)~&=~{\frac {\ \sin {\sqrt {x\ }}\ }{\sqrt {x\ }}}~.\end{aligned}}}

<p>Similarly, by comparing with the expansion of the <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic functions</a> sinh and cosh we find for 
  
    
      
         
        x
        <
        0
         
        :
      
    
    {\displaystyle ~x<0\ :}

</p>

c
                  
                    0
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                 
                cosh
                ⁡
                
                  
                    −
                    x
                     
                  
                
                 
                ,
              
            
            
              
                
                  c
                  
                    1
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                
                  
                    
                       
                      sinh
                      ⁡
                      
                        
                          −
                          x
                           
                        
                      
                       
                    
                    
                      −
                      x
                       
                    
                  
                
                 
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}c_{0}(x)~&=~~\cosh {\sqrt {-x\ }}\ ,\\[1ex]c_{1}(x)~&=~{\frac {\ \sinh {\sqrt {-x\ }}\ }{\sqrt {-x\ }}}~.\end{aligned}}}

<p>Circular orbits and elliptical orbits use sine and cosine relations, and hyperbolic orbits use the sinh and cosh relations. Parabolic orbits (marginal escape orbits) formulas are a special in-between case. 
</p>
<h2 id="recursion">Recursion</h2>
<p>For higher-order Stumpff functions needed for both ordinary trajectories and for <a href="/facts/Perturbation_theory/JWrNgbVm">perturbation theory</a>, one can use the <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a>:
</p>

x
         
        
          c
          
            k
            +
            2
          
        
        (
        x
        )
        =
        
          
            1
            
              k
              !
            
          
        
        −
        
          c
          
            k
          
        
        (
        x
        )
         
      
    
    {\displaystyle \ x\ c_{k+2}(x)={\frac {1}{k!}}-c_{k}(x)~}
  
 for 
  
    
      
         
        k
        =
        0
        ,
        1
        ,
        2
        ,
         
        …
         
         
        ,
      
    
    {\displaystyle ~k=0,1,2,\ \ldots \ ~,}

<p>or when 
  
    
      
         
        x
        ≠
        0
         
      
    
    {\displaystyle \ x\neq 0\ }

</p>

c
          
            k
            +
            2
          
        
        (
        x
        )
         
        =
         
        
          
            
               
              1
               
            
            x
          
        
        
          (
          
             
            
              
                1
                
                  k
                  !
                
              
            
            −
            
              c
              
                k
              
            
            (
            x
            )
             
          
          )
        
         
      
    
    {\displaystyle \ c_{k+2}(x)~=~{\frac {\ 1\ }{x}}\left(\ {\frac {1}{k!}}-c_{k}(x)\ \right)~}
  
 for 
  
    
      
         
        k
        =
        0
        ,
        1
        ,
        2
        ,
         
        …
         
         
        .
      
    
    {\displaystyle ~k=0,1,2,\ \ldots ~~.}

<p>Using this recursion, the two further Stumpf functions needed for the <a href="/facts/Universal_variable_formulation/nXGMrHA8">universal variable formulation</a> are, for 
  
    
      
         
        x
        >
        0
         
        :
      
    
    {\displaystyle ~x>0\ :}

</p>

c
                  
                    2
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                 
                 
                
                  
                    
                       
                      1
                      −
                      cos
                      ⁡
                      
                        
                          x
                           
                        
                      
                       
                    
                    
                      x
                       
                    
                  
                
                 
                ,
              
            
            
              
            
            
              
                 
                
                  c
                  
                    3
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                
                  
                    
                       
                      
                        
                          x
                           
                        
                      
                      −
                      sin
                      ⁡
                      
                        
                          x
                           
                        
                      
                       
                    
                    x
                  
                
                 
                ;
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\ c_{2}(x)~&=~~~{\frac {\ 1-\cos {\sqrt {x\ }}\ }{\sqrt {x\ }}}\ ,\\\\\ c_{3}(x)~&=~{\frac {\ {\sqrt {x\ }}-\sin {\sqrt {x\ }}\ }{x}}\ ;\end{aligned}}}

<p>and for 
  
    
      
         
        x
        <
        0
         
        :
      
    
    {\displaystyle ~x<0\ :}

</p>

c
                  
                    2
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                
                
                  
                    
                       
                      1
                      −
                      cosh
                      ⁡
                      
                        
                          −
                          x
                           
                        
                      
                       
                    
                    
                      −
                      x
                       
                    
                  
                
                 
                ,
              
            
            
              
            
            
              
                 
                
                  c
                  
                    3
                  
                
                (
                x
                )
                 
              
              
                
                =
                 
                
                  
                    
                       
                      
                        
                          −
                          x
                           
                        
                      
                       
                      −
                       
                      sinh
                      ⁡
                      
                        
                          −
                          x
                           
                        
                      
                       
                    
                    
                      −
                      x
                    
                  
                
                 
                 
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\ c_{2}(x)~&=~\quad {\frac {\ 1-\cosh {\sqrt {-x\ }}\ }{\sqrt {-x\ }}}\ ,\\\\\ c_{3}(x)~&=~{\frac {\ {\sqrt {-x\ }}\ -\ \sinh {\sqrt {-x\ }}\ }{-x}}~~.\end{aligned}}}

<h2 id="relations-to-other-functions">Relations to other functions</h2>
<p>The Stumpff functions can be expressed in terms of the <a href="/facts/Mittag-Leffler_function/qWjX1gP2">Mittag-Leffler function</a>:
</p>

c
          
            k
          
        
        (
        x
        )
         
        =
         
        
          E
          
            2
            ,
            k
            +
            1
          
        
        (
        −
        x
        )
         
         
        .
      
    
    {\displaystyle \ c_{k}(x)~=~E_{2,k+1}(-x)~~.}

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Karl Stumpff (1956). Himmelsmechanik [Celestial Mechanics] (in German). Deutscher Verlag der Wissenschaften. <a href="/wiki/Karl_Stumpff" target="_blank">/wiki/Karl_Stumpff</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Danby, J.M.A. (1988). Fundamentals of Celestial Mechanics (revised ed.). Willman–Bell. ISBN 9780023271403. <a href="9780023271403" target="_blank">9780023271403</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Stiefel, Eduard; Scheifele, Gerhard (1971). Linear and Regular Celestial Mechanics: Perturbed two-body motion, numerical methods, canonical theory. Springer-Verlag. ISBN 978-0-38705119-2. <a href="978-0-38705119-2" target="_blank">978-0-38705119-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Stumpff function open-in-new

Stumpff function