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''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" />
+
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 $
  
<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>
+
$$ \tag{* }
 +
F (w)  = \
 +
f (a) +
 +
$$
  
<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>
+
$$
 +
+
 +
\sum _ {n = 1 } ^  \infty  {
 +
\frac{1}{n!}
 +
} \left \{
 +
\frac{d ^ {n - 1 } }{dz ^ {n - 1 } }
  
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" />.
+
\left [ f ^ { \prime } (z) \left (
 +
\frac{z - a }{g (z) - b }
  
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
+
\right ) ^ {n}  \right ] \right \} _ {z = a }  (w - b)  ^ {n} .
 +
$$
  
<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>
+
The inverse of the function  $  w = g(z) $
 +
is obtained by setting  $  f(z) \equiv z $.
  
<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>
+
The expansion (*) follows from Bürmann's theorem [[#References|[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
 
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>
+
$$
 +
R _ {m}  =
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _ { a } ^ { z }  \int\limits _  \gamma
 +
\left [
 +
 
 +
\frac{g (z) - b }{g (t) - b }
  
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" />.
+
\right ] ^ {m - 1 }
  
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 .
+
\frac{f ^ { \prime } (t) g  ^  \prime  (z)  dt  dz }{g (t) - g (z) }
 +
.
 +
$$
  
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]]]:
+
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 $.
  
<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>
+
The expansion (*) for the case  $  b = 0 $
 +
was obtained by J.L. Lagrange .
  
<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>
+
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 [[#References|[3]]]:
  
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" />.
+
$$
 +
F (w) = f (a) +
 +
$$
 +
 
 +
$$
 +
+
 +
\sum _ {n = 1 } ^  \infty  {
 +
\frac{1}{n!}
 +
} \left \{
 +
\frac{d ^ {n - 1 } }{dz ^ {n - 1 } }
 +
\left [ f ^
 +
{ \prime } (z) \left (
 +
\frac{z - a }{g (z) - b }
 +
\right )
 +
^ {n}  \right ]  \right \} _ {z = a }  (w - b)  ^ {n/r} .
 +
$$
 +
 
 +
Another generalization (see, for example, [[#References|[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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 06:29, 30 May 2020


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)
How to Cite This Entry:
Bürmann-Lagrange series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B%C3%BCrmann-Lagrange_series&oldid=22221
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article