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To obtain, for a given power series
 
To obtain, for a given power series
  
<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/i/i052/i052450/i0524501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
w  =  f ( z)  = \
 +
\sum_{j=0} ^  \infty 
 +
b _ {j} ( z- a )  ^ {j} ,\ \
 +
b _ {1} \neq 0 ,
 +
$$
 +
 
 +
a series for the inverse function  $  z = \phi ( w) $
 +
in the form
 +
 
 +
$$ \tag{2 }
 +
= \phi ( w)  = \
 +
\sum_{k=0} ^  \infty 
 +
a _ {k} ( w - b )  ^ {k} ,
 +
$$
 +
 
 +
where  $  b = f ( a) = b _ {0} $,
 +
$  a _ {0} = a $,
 +
 
 +
$$
 +
a _ {k}  = \
 +
 
 +
\frac{1}{k!}
 +
\
 +
\lim\limits _ {\zeta \rightarrow a } \
  
a series for the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524502.png" /> in the form
+
\frac{d  ^ {k-1} }{d \zeta  ^ {k-1} }
  
<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/i/i052/i052450/i0524503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\left [
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524505.png" />,
+
\frac{\zeta - a }{f ( \zeta ) - 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/i/i052/i052450/i0524506.png" /></td> </tr></table>
+
\right ]  ^ {k} ,\ \
 +
k \geq  1 .
 +
$$
  
The series (2) is called the inverse of the series (1), or the Lagrange series. The more general problem of finding the expansion of an arbitrary composite analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524507.png" /> is solved by the [[Bürmann–Lagrange series|Bürmann–Lagrange series]]. If the disc of convergence of (1) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524508.png" />, then the series (2) converges in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i0524509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245010.png" /> is the distance of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245011.png" /> from the image of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245012.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245013.png" />.
+
The series (2) is called the inverse of the series (1), or the Lagrange series. The more general problem of finding the expansion of an arbitrary composite analytic function $  F [ \phi ( w) ] $
 +
is solved by the [[Bürmann–Lagrange series|Bürmann–Lagrange series]]. If the disc of convergence of (1) is $  | z- a | < \rho $,  
 +
then the series (2) converges in the disc $  | w- b | < \delta $,  
 +
where $  \delta $
 +
is the distance of the point $  b $
 +
from the image of the circle $  | z- a | = \rho $
 +
under the mapping $  w = f ( z) $.
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245014.png" /> is expanded as a series of the form
+
If the function $  w = f ( z) $
 +
is expanded as a series of 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/i/i052/i052450/i05245015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
= f ( z)  = \
 +
b + \sum_{j=m} ^  \infty 
 +
b _ {j} ( z- a )  ^ {j} ,\ \
 +
m \geq  2 ,\ \
 +
b _ {m} \neq 0 ,
 +
$$
  
that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245016.png" /> is a [[Critical point|critical point]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245017.png" />, then the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245018.png" /> has an [[Algebraic branch point|algebraic branch point]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245020.png" />, and inversion of (3) is only possible in the form of a [[Puiseux series|Puiseux series]]:
+
that is, if $  a $
 +
is a [[critical point]] for $  f ( z) $,  
 +
then the inverse function $  z = \phi ( w) $
 +
has an [[Algebraic branch point|algebraic branch point]] of order $  m- 1 $
 +
at $  b $,  
 +
and inversion of (3) is only possible in the form of a [[Puiseux series|Puiseux series]]:
  
<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/i/i052/i052450/i05245021.png" /></td> </tr></table>
+
$$
 +
= \phi ( w)  = \
 +
a + \sum_{k=1} ^  \infty 
 +
a _ {k} ( w- b ) ^ {k / m } ,
 +
$$
  
<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/i/i052/i052450/i05245022.png" /></td> </tr></table>
+
$$
 +
a _ {k}  =
 +
\frac{1}{k ! }
 +
  \lim\limits _ {\zeta
 +
\rightarrow a } 
 +
\frac{d  ^ {k-1} }{d \zeta  ^ {k-1} }
 +
\left \{
 +
\frac{\zeta
 +
- a }{[ f ( \zeta ) - b ] ^ {1 / m } }
 +
\right \}  ^ {k} ,\  k \geq  1 .
 +
$$
  
The problem of inversion of a [[Laurent series|Laurent series]] in negative and positive integer powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245023.png" /> is solved similarly in the case when the series has only finitely many negative (or positive) powers (see [[#References|[1]]]).
+
The problem of inversion of a [[Laurent series]] in negative and positive integer powers of $  z- a $
 +
is solved similarly in the case when the series has only finitely many negative (or positive) powers (see [[#References|[1]]]).
  
For analytic functions of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245025.png" />, problems of inversion can be put in various ways. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245026.png" /> is a non-singular (that is, the rank of the Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245027.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245028.png" />) holomorphic mapping of a neighbourhood of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245031.png" />, then in some neighbourhood of zero there exists an inverse holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052450/i05245032.png" />, which can be described in the form of a multi-dimensional Bürmann–Lagrange series (see [[#References|[3]]]).
+
For analytic functions of several complex variables $  z = ( z _ {1} \dots z _ {n} ) $,  
 +
$  n > 1 $,  
 +
problems of inversion can be put in various ways. For example, if $  f : \mathbf C  ^ {n} \rightarrow \mathbf C  ^ {n} $
 +
is a non-singular (that is, the rank of the Jacobi matrix $  \| \partial  f / \partial  z _ {k} \| $
 +
is equal to $  n $)  
 +
holomorphic mapping of a neighbourhood of zero in $  \mathbf C  ^ {n} $
 +
into $  \mathbf C  ^ {n} $,
 +
$  f ( 0) = 0 $,  
 +
then in some neighbourhood of zero there exists an inverse holomorphic function $  \phi $,  
 +
which can be described in the form of a multi-dimensional Bürmann–Lagrange series (see [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer  (1964)  pp. Chapt. 3, Abschnitt 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.E. Soltan,  "The expansion of holomorphic functions in multi-dimensional Bürmann–Lagrange series" , ''Holomorphic functions of several complex variables'' , Krasnoyarsk  (1972)  pp. 129–137; 212  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer  (1964)  pp. Chapt. 3, Abschnitt 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.E. Soltan,  "The expansion of holomorphic functions in multi-dimensional Bürmann–Lagrange series" , ''Holomorphic functions of several complex variables'' , Krasnoyarsk  (1972)  pp. 129–137; 212  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 20:40, 16 January 2024


To obtain, for a given power series

$$ \tag{1 } w = f ( z) = \ \sum_{j=0} ^ \infty b _ {j} ( z- a ) ^ {j} ,\ \ b _ {1} \neq 0 , $$

a series for the inverse function $ z = \phi ( w) $ in the form

$$ \tag{2 } z = \phi ( w) = \ \sum_{k=0} ^ \infty a _ {k} ( w - b ) ^ {k} , $$

where $ b = f ( a) = b _ {0} $, $ a _ {0} = a $,

$$ a _ {k} = \ \frac{1}{k!} \ \lim\limits _ {\zeta \rightarrow a } \ \frac{d ^ {k-1} }{d \zeta ^ {k-1} } \left [ \frac{\zeta - a }{f ( \zeta ) - b } \right ] ^ {k} ,\ \ k \geq 1 . $$

The series (2) is called the inverse of the series (1), or the Lagrange series. The more general problem of finding the expansion of an arbitrary composite analytic function $ F [ \phi ( w) ] $ is solved by the Bürmann–Lagrange series. If the disc of convergence of (1) is $ | z- a | < \rho $, then the series (2) converges in the disc $ | w- b | < \delta $, where $ \delta $ is the distance of the point $ b $ from the image of the circle $ | z- a | = \rho $ under the mapping $ w = f ( z) $.

If the function $ w = f ( z) $ is expanded as a series of the form

$$ \tag{3 } w = f ( z) = \ b + \sum_{j=m} ^ \infty b _ {j} ( z- a ) ^ {j} ,\ \ m \geq 2 ,\ \ b _ {m} \neq 0 , $$

that is, if $ a $ is a critical point for $ f ( z) $, then the inverse function $ z = \phi ( w) $ has an algebraic branch point of order $ m- 1 $ at $ b $, and inversion of (3) is only possible in the form of a Puiseux series:

$$ z = \phi ( w) = \ a + \sum_{k=1} ^ \infty a _ {k} ( w- b ) ^ {k / m } , $$

$$ a _ {k} = \frac{1}{k ! } \lim\limits _ {\zeta \rightarrow a } \frac{d ^ {k-1} }{d \zeta ^ {k-1} } \left \{ \frac{\zeta - a }{[ f ( \zeta ) - b ] ^ {1 / m } } \right \} ^ {k} ,\ k \geq 1 . $$

The problem of inversion of a Laurent series in negative and positive integer powers of $ z- a $ is solved similarly in the case when the series has only finitely many negative (or positive) powers (see [1]).

For analytic functions of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n > 1 $, problems of inversion can be put in various ways. For example, if $ f : \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ is a non-singular (that is, the rank of the Jacobi matrix $ \| \partial f / \partial z _ {k} \| $ is equal to $ n $) holomorphic mapping of a neighbourhood of zero in $ \mathbf C ^ {n} $ into $ \mathbf C ^ {n} $, $ f ( 0) = 0 $, then in some neighbourhood of zero there exists an inverse holomorphic function $ \phi $, which can be described in the form of a multi-dimensional Bürmann–Lagrange series (see [3]).

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2
[3] E.E. Soltan, "The expansion of holomorphic functions in multi-dimensional Bürmann–Lagrange series" , Holomorphic functions of several complex variables , Krasnoyarsk (1972) pp. 129–137; 212 (In Russian)
How to Cite This Entry:
Inversion of a series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inversion_of_a_series&oldid=16637
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article