Lagrange reversion theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Lagrange reversion theorem gives <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> or <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> expansions of certain <a href="/facts/Implicitly_defined_function/UJmQ2Cyk">implicitly defined functions</a>; 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 which case the equation can be derived using <a href="/facts/Perturbation_theory/JWrNgbVm">perturbation theory</a>.
In 1770, <a href="/facts/Joseph_Louis_Lagrange/bTQUaSTv">Joseph Louis Lagrange</a> (1736–1813) published his <a href="/facts/Power_series/vYh6sTps">power series</a> solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, <a href="/facts/Pierre-Simon_Laplace/6ktp0txb">Pierre-Simon Laplace</a> (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. <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a> (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 <a href="/facts/Kepler%2527s_equation/pww75aK5">Kepler's equation</a>.

Lagrange reversion theorem open-in-new

Lagrange reversion theorem