Bürmann-Lagrange series

Lagrange series

A power series which offers a complete solution to the problem of local inversion of holomorphic functions. In fact, let a function $w = g(z)$ of the complex variable $z$ be regular in a neighbourhood of the point $z = a$, and let $g(a) = b$ and $g ^ \prime (a) \neq 0$. Then there exists a regular function $z = h (w)$ in some neighbourhood of the point $w = b$ of the $w$- plane which is the inverse to $g(z)$ and is such that $h(b) = a$. Moreover, if $f(z)$ is any regular function in a neighbourhood of the point $z = a$, then the composite function $F(w) = f[h(w)]$ can be expanded in a Bürmann–Lagrange series in a neighbourhood of the point $w = b$

$$\tag{* } F (w) = \ f (a) +$$

$$+ \sum _ {n = 1 } ^ \infty { \frac{1}{n!} } \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n} .$$

The inverse of the function $w = g(z)$ is obtained by setting $f(z) \equiv z$.

The expansion (*) follows from Bürmann's theorem [1]: Under the assumptions made above on the holomorphic functions $g(z)$ and $f(z)$, the latter function may be represented in a certain domain in the $z$- plane containing $a$ in the form

$$f (z) = f (a) +$$

$$+ \sum _ {n = 1 } ^ { {m } - 1 } \frac{[g (z) - b] ^ {n} }{n!} \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } + R _ {m} ,$$

where

$$R _ {m} = \frac{1}{2 \pi i } \int\limits _ { a } ^ { z } \int\limits _ \gamma \left [ \frac{g (z) - b }{g (t) - b } \right ] ^ {m - 1 } \frac{f ^ { \prime } (t) g ^ \prime (z) dt dz }{g (t) - g (z) } .$$

Here $\gamma$ is a contour in the $t$- plane which encloses the points $a$ and $z$, and is such that if $\zeta$ is any point inside $\gamma$, then the equation $g(t) = g( \zeta )$ has no roots on $\gamma$ or inside $\gamma$ other than the simple root $t = \zeta$.

The expansion (*) for the case $b = 0$ was obtained by J.L. Lagrange .

If the derivative $g ^ \prime (t)$ has a zero of order $r - 1$ at the point $z = a$, there is the following generalization of the Bürmann–Lagrange series for the multi-valued inverse function [3]:

$$F (w) = f (a) +$$

$$+ \sum _ {n = 1 } ^ \infty { \frac{1}{n!} } \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n/r} .$$

Another generalization (see, for example, [4]) refers to functions $g(z)$ regular in an annulus; instead of the series (*), one obtains a series with positive and negative powers of the difference $w - b$.

References

 [1] 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 [3] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1968) pp. Chapt. 7 [4] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 [5] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)