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''algebraic singular point''
 
''algebraic singular point''
  
An isolated branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114301.png" /> of finite order of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114302.png" />, having the property that the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114303.png" /> exists for any regular element of continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114304.png" /> in a domain for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114305.png" /> is a boundary point. More exactly, a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114306.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114307.png" />-plane for the complete [[Analytic function|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114308.png" />, under continuation of some regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a0114309.png" /> of this function with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143010.png" /> along paths passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143011.png" />, is called an algebraic branch point if it fulfills the following conditions: 1) There exists a positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143012.png" /> such that the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143013.png" /> may be extended along an arbitrary continuous curve lying in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143014.png" />; 2) there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143015.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143016.png" /> is an arbitrary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143017.png" />, the analytic continuation of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143019.png" /> yields exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143020.png" /> different elements of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143021.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143022.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143023.png" /> is an arbitrary element with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143024.png" />, all the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143025.png" /> elements with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143026.png" /> can be obtained by analytic continuation along closed paths around the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143027.png" />; and 3) the values at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143029.png" /> of all elements which are obtainable from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143030.png" /> by continuation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143031.png" /> tend to a definite, finite or infinite, limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143032.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143033.png" /> while remaining in D.
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An isolated branch point $a$ of finite order of an analytic function $f(z)$, having the property that the limit $\lim_{z\to a}f(z)$ exists for any regular element of continuation of $f$ in a domain for which $a$ is a boundary point. More exactly, a singular point $a$ in the complex $z$-plane for the complete [[Analytic function|analytic function]] $f(z)$, under continuation of some regular element $e_0$ of this function with centre $z_0$ along paths passing through $a$, is called an algebraic branch point if it fulfills the following conditions: 1) There exists a positive number $\rho$ such that the element $e_0$ may be extended along an arbitrary continuous curve lying in the annulus $D=\{z:0<|z-a|<\rho\}$; 2) there exists a positive integer $k>1$ such that if $z_1$ is an arbitrary point of $D$, the analytic continuation of the element $e_0$ in $D$ yields exactly $k$ different elements of the function $f(z)$ with centre $z_1$; if $e_1$ is an arbitrary element with centre $z_1$, all the remaining $k-1$ elements with centre $z_1$ can be obtained by analytic continuation along closed paths around the point $a$; and 3) the values at the points $z$ of $D$ of all elements which are obtainable from $e_0$ by continuation in $D$ tend to a definite, finite or infinite, limit as $z$ tends to $a$ while remaining in $D$.
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143034.png" /> is said to be the order of the algebraic branch point. All branches of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143035.png" /> obtainable by analytic continuation of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143036.png" /> in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143037.png" /> may be represented in a deleted neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143038.png" /> by a generalized Laurent series (Puiseux series):
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The number $k-1$ is said to be the order of the algebraic branch point. All branches of the function $f(z)$ obtainable by analytic continuation of the element $e_0$ in the annulus $D$ may be represented in a deleted neighbourhood of $a$ by a generalized Laurent 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/a/a011/a011430/a01143039.png" /></td> </tr></table>
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$$f(z)=\sum_{n=-m}^\infty c_n(z-a)^{n/k},\quad m\geq0.$$
  
The point at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143040.png" />, is called an algebraic branch point for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143041.png" /> if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143042.png" /> is an algebraic branch point of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143043.png" />.
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The point at infinity, $a=\infty$, is called an algebraic branch point for a function $f(z)$ if the point $b=0$ is an algebraic branch point of the function $g(w)=f(1/w)$.
  
There may exist several (and even an infinite number of) different algebraic branch points and regular points of a complete analytic function with a given affix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011430/a01143044.png" />.
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There may exist several (and even an infinite number of) different algebraic branch points and regular points of a complete analytic function with a given affix $a$.
  
 
====References====
 
====References====
 
<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><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''4''' , Springer  (1968)</TD></TR></table>
 
<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><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''4''' , Springer  (1968)</TD></TR></table>

Latest revision as of 15:19, 29 December 2018

algebraic singular point

An isolated branch point $a$ of finite order of an analytic function $f(z)$, having the property that the limit $\lim_{z\to a}f(z)$ exists for any regular element of continuation of $f$ in a domain for which $a$ is a boundary point. More exactly, a singular point $a$ in the complex $z$-plane for the complete analytic function $f(z)$, under continuation of some regular element $e_0$ of this function with centre $z_0$ along paths passing through $a$, is called an algebraic branch point if it fulfills the following conditions: 1) There exists a positive number $\rho$ such that the element $e_0$ may be extended along an arbitrary continuous curve lying in the annulus $D=\{z:0<|z-a|<\rho\}$; 2) there exists a positive integer $k>1$ such that if $z_1$ is an arbitrary point of $D$, the analytic continuation of the element $e_0$ in $D$ yields exactly $k$ different elements of the function $f(z)$ with centre $z_1$; if $e_1$ is an arbitrary element with centre $z_1$, all the remaining $k-1$ elements with centre $z_1$ can be obtained by analytic continuation along closed paths around the point $a$; and 3) the values at the points $z$ of $D$ of all elements which are obtainable from $e_0$ by continuation in $D$ tend to a definite, finite or infinite, limit as $z$ tends to $a$ while remaining in $D$.

The number $k-1$ is said to be the order of the algebraic branch point. All branches of the function $f(z)$ obtainable by analytic continuation of the element $e_0$ in the annulus $D$ may be represented in a deleted neighbourhood of $a$ by a generalized Laurent series (Puiseux series):

$$f(z)=\sum_{n=-m}^\infty c_n(z-a)^{n/k},\quad m\geq0.$$

The point at infinity, $a=\infty$, is called an algebraic branch point for a function $f(z)$ if the point $b=0$ is an algebraic branch point of the function $g(w)=f(1/w)$.

There may exist several (and even an infinite number of) different algebraic branch points and regular points of a complete analytic function with a given affix $a$.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt.8 (Translated from Russian)
[2] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 4 , Springer (1968)
How to Cite This Entry:
Algebraic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_branch_point&oldid=14573
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article