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
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 .
|||H. Bürmann, Mem. Inst. Nat. Sci. Arts. Sci. Math. Phys. , 2 (1799) pp. 13–17|
|[2a]||J.L. Lagrange, Mem. Acad. R. Sci. et Belles-lettres Berlin , 24 (1770)|
|[2b]||J.L. Lagrange, "Additions au mémoire sur la résolution des équations numériques" , Oeuvres , 2 , G. Olms (1973) pp. 579–652|
|||A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1968) pp. Chapt. 7|
|||E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6|
|||A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)|
There is an exhaustive treatment of the Lagrange–Bürmann theorem and series in [a1].
|[a1]||P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (1974)|
Bürmann–Lagrange series. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=B%C3%BCrmann%E2%80%93Lagrange_series&oldid=23202