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Difference between revisions of "Analytic image"

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An extension of the concept of a [[Complete analytic function|complete analytic function]], obtained on considering all possible elements of an analytic function in the form of generalized power series (Puiseux series)
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An extension of the concept of a [[complete analytic function]], obtained on considering all possible elements of an analytic function in the form of generalized power series ([[Puiseux series]])
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$$
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\begin{equation}\label{*}
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\sum_{\nu=m}^\infty a_\nu (z-z_0)^{\nu/n}\ , \ \ \ \sum_{\nu=m}^\infty a_\nu z^{-\nu/n}
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\end{equation}
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$$
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Here $z$ is a complex variable, $m$ is an integer and $n$ is a natural number. The series converge in the domains $|z-z_0| < r$ and $|z| > r > 0$, respectively. An analytic image can be identified with the class of all elements of the form \eqref{*} which are obtained from each other by [[analytic continuation]]. The analytic image differs from the complete analytic function by the addition of all ramified elements of the form \eqref{*} with $n>1$, which are obtained by analytic continuation of its regular elements with $n=1$. After the introduction of a suitable topology, the analytic image is converted to the [[Riemann surface]] of the given function.
  
<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/a/a012/a012300/a0123001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 8  (Translated from Russian)</TD></TR>
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</table>
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123002.png" /> is a complex variable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123003.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123004.png" /> is a natural number. The series converge in the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123006.png" />, respectively. An analytic image can be identified with the class of all elements of the form (*) which are obtained from each other by [[Analytic continuation|analytic continuation]]. The analytic image differs from the complete analytic function by the addition of all ramified elements of the form (*) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123007.png" />, which are obtained by analytic continuation of its regular elements with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012300/a0123008.png" />. After the introduction of a suitable topology, the analytic image is converted to the [[Riemann surface|Riemann surface]] of the given function.
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{{TEX|done}}
  
====References====
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[[Category:Functions of a complex variable]]
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 8  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 17:44, 1 November 2014

An extension of the concept of a complete analytic function, obtained on considering all possible elements of an analytic function in the form of generalized power series (Puiseux series) $$ \begin{equation}\label{*} \sum_{\nu=m}^\infty a_\nu (z-z_0)^{\nu/n}\ , \ \ \ \sum_{\nu=m}^\infty a_\nu z^{-\nu/n} \end{equation} $$ Here $z$ is a complex variable, $m$ is an integer and $n$ is a natural number. The series converge in the domains $|z-z_0| < r$ and $|z| > r > 0$, respectively. An analytic image can be identified with the class of all elements of the form \eqref{*} which are obtained from each other by analytic continuation. The analytic image differs from the complete analytic function by the addition of all ramified elements of the form \eqref{*} with $n>1$, which are obtained by analytic continuation of its regular elements with $n=1$. After the introduction of a suitable topology, the analytic image is converted to the Riemann surface of the given function.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)
How to Cite This Entry:
Analytic image. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_image&oldid=18574
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article