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)

Comments

There is an exhaustive treatment of the Lagrange–Bürmann theorem and series in [a1].

References

[a1]	P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (1974)