Lagrange reversion theorem | Wikipedia audio article
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Date: 31/12/2018

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SUMMARY
=======
This page is about Lagrange reversion. For inversion, see Lagrange inversion theorem.In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions.
Let v be a function of x and y in terms of another function f such that

  
    
      
        v
        =
        x
        +
        y
        f
        (
        v
        )
      
    
    {\displaystyle v=x+yf(v)}
  Then for any function g, for small enough y:

  
    
      
        g
        (
        v
        )
        =
        g
        (
        x
        )
        +
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              y
              
                k
              
            
            
              k
              !
            
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  x
                
              
            
            )
          
          
            k
            −
            1
          
        
        
          (
          
            f
            (
            x
            
              )
              
                k
              
            
            
              g
              ′
            
            (
            x
            )
          
          )
        
        .
      
    
    {\displaystyle g(v)=g(x)+\sum _{k=1}^{\infty }{\frac {y^{k}}{k!}}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}g'(x)\right).}
  If g is the identity, this becomes

  
    
      
        v
        =
        x
        +
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              y
              
                k
              
            
            
              k
              !
            
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  x
                
              
            
            )
          
          
            k
            −
            1
          
        
        
          (
          
            f
            (
            x
            
              )
              
                k
              
            
          
          )
        
        .
      
    
    {\displaystyle v=x+\sum _{k=1}^{\infty }{\frac {y^{k}}{k!}}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}\right).}
  In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation.