Inversion of a series

To obtain, for a given power series

$$ \tag{1 } w = f ( z) = \ \sum_{j=0} ^ \infty b _ {j} ( z- a ) ^ {j} ,\ \ b _ {1} \neq 0 , $$

a series for the inverse function $ z = \phi ( w) $ in the form

$$ \tag{2 } z = \phi ( w) = \ \sum_{k=0} ^ \infty a _ {k} ( w - b ) ^ {k} , $$

where $ b = f ( a) = b _ {0} $, $ a _ {0} = a $,

$$ a _ {k} = \ \frac{1}{k!} \ \lim\limits _ {\zeta \rightarrow a } \ \frac{d ^ {k-} 1 }{d \zeta ^ {k-} 1 } \left [ \frac{\zeta - a }{f ( \zeta ) - b } \right ] ^ {k} ,\ \ k \geq 1 . $$

The series (2) is called the inverse of the series (1), or the Lagrange series. The more general problem of finding the expansion of an arbitrary composite analytic function $ F [ \phi ( w) ] $ is solved by the Bürmann–Lagrange series. If the disc of convergence of (1) is $ | z- a | < \rho $, then the series (2) converges in the disc $ | w- b | < \delta $, where $ \delta $ is the distance of the point $ b $ from the image of the circle $ | z- a | = \rho $ under the mapping $ w = f ( z) $.

If the function $ w = f ( z) $ is expanded as a series of the form

$$ \tag{3 } w = f ( z) = \ b + \sum_{j=m} ^ \infty b _ {j} ( z- a ) ^ {j} ,\ \ m \geq 2 ,\ \ b _ {m} \neq 0 , $$

that is, if $ a $ is a critical point for $ f ( z) $, then the inverse function $ z = \phi ( w) $ has an algebraic branch point of order $ m- 1 $ at $ b $, and inversion of (3) is only possible in the form of a Puiseux series:

$$ z = \phi ( w) = \ a + \sum_{k=1} ^ \infty a _ {k} ( w- b ) ^ {k / m } , $$

$$ a _ {k} = \frac{1}{k ! } \lim\limits _ {\zeta \rightarrow a } \frac{d ^ {k-} 1 }{d \zeta ^ {k-} 1 } \left \{ \frac{\zeta - a }{[ f ( \zeta ) - b ] ^ {1 / m } } \right \} ^ {k} ,\ k \geq 1 . $$

The problem of inversion of a Laurent series in negative and positive integer powers of $ z- a $ is solved similarly in the case when the series has only finitely many negative (or positive) powers (see [1]).

For analytic functions of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n > 1 $, problems of inversion can be put in various ways. For example, if $ f : \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ is a non-singular (that is, the rank of the Jacobi matrix $ \| \partial f / \partial z _ {k} \| $ is equal to $ n $) holomorphic mapping of a neighbourhood of zero in $ \mathbf C ^ {n} $ into $ \mathbf C ^ {n} $, $ f ( 0) = 0 $, then in some neighbourhood of zero there exists an inverse holomorphic function $ \phi $, which can be described in the form of a multi-dimensional Bürmann–Lagrange series (see [3]).

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2
[3]	E.E. Soltan, "The expansion of holomorphic functions in multi-dimensional Bürmann–Lagrange series" , Holomorphic functions of several complex variables , Krasnoyarsk (1972) pp. 129–137; 212 (In Russian)