# Bürmann-Lagrange series

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Lagrange series

A power series which offers a complete solution to the problem of local inversion of holomorphic functions. In fact, let a function of the complex variable be regular in a neighbourhood of the point , and let and . Then there exists a regular function in some neighbourhood of the point of the -plane which is the inverse to and is such that . Moreover, if is any regular function in a neighbourhood of the point , then the composite function can be expanded in a Bürmann–Lagrange series in a neighbourhood of the point  (*) The inverse of the function is obtained by setting .

The expansion (*) follows from Bürmann's theorem : Under the assumptions made above on the holomorphic functions and , the latter function may be represented in a certain domain in the -plane containing in the form  where Here is a contour in the -plane which encloses the points and , and is such that if is any point inside , then the equation has no roots on or inside other than the simple root .

The expansion (*) for the case was obtained by J.L. Lagrange .

If the derivative has a zero of order at the point , there is the following generalization of the Bürmann–Lagrange series for the multi-valued inverse function :  Another generalization (see, for example, ) refers to functions regular in an annulus; instead of the series (*), one obtains a series with positive and negative powers of the difference .