View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177901.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177902.png" /> be regular in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177903.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177905.png" />. Then there exists a regular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177906.png" /> in some neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177907.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177908.png" />-plane which is the inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b0177909.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779010.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779011.png" /> is any regular function in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779012.png" />, then the composite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779013.png" /> can be expanded in a Bürmann–Lagrange series in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779014.png" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779016.png" /></td> </tr></table>

The inverse of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779017.png" /> is obtained by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779018.png" />.

The expansion (*) follows from Bürmann's theorem [[#References|[1]]]: Under the assumptions made above on the holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779020.png" />, the latter function may be represented in a certain domain in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779021.png" />-plane containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779022.png" /> in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779023.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779024.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779025.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779026.png" /> is a contour in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779027.png" />-plane which encloses the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779029.png" />, and is such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779030.png" /> is any point inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779031.png" />, then the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779032.png" /> has no roots on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779033.png" /> or inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779034.png" /> other than the simple root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779035.png" />.

The expansion (*) for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779036.png" /> was obtained by J.L. Lagrange .

If the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779037.png" /> has a zero of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779038.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779039.png" />, there is the following generalization of the Bürmann–Lagrange series for the multi-valued inverse function [[#References|[3]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779040.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779041.png" /></td> </tr></table>

Another generalization (see, for example, [[#References|[4]]]) refers to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779042.png" /> regular in an annulus; instead of the series (*), one obtains a series with positive and negative powers of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017790/b01779043.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bürmann,   ''Mem. Inst. Nat. Sci. Arts. Sci. Math. Phys.'' , '''2'''  (1799)  pp. 13–17</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  J.L. Lagrange,   ''Mem. Acad. R. Sci. et Belles-lettres Berlin'' , '''24'''  (1770)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  J.L. Lagrange,   "Additions au mémoire sur la résolution des équations numériques" , ''Oeuvres'' , '''2''' , G. Olms  (1973)  pp. 579–652</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer  (1968)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====
There is an exhaustive treatment of the Lagrange–Bürmann theorem and series in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Henrici,   "Applied and computational complex analysis" , '''1''' , Wiley  (1974)</TD></TR></table>