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''branch point of infinite order''
 
''branch point of infinite order''
  
A special form of a [[Branch point|branch point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605601.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605602.png" /> of one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605603.png" />, when for no finite number of successive circuits in the same direction about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605604.png" /> the [[Analytic continuation|analytic continuation]] of some element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605605.png" /> returns to the original element. More precisely, an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605606.png" /> is called a logarithmic branch point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605607.png" /> if there exist: 1) an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605608.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605609.png" /> can be analytically continued along any path; and 2) a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056010.png" /> and an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056011.png" /> in the form of a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056012.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056013.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056014.png" />, the analytic continuation of which along the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056015.png" />, taken arbitrarily many times in the same direction, never returns to the original element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056016.png" />. In the case of a logarithmic branch point at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056017.png" />, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056018.png" /> one must consider a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056019.png" />. Logarithmic branch points belong to the class of transcendental branch points (cf. [[Transcendental branch point|Transcendental branch point]]). The behaviour of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056020.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056021.png" /> in the presence of a logarithmic branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056022.png" /> is characterized by the fact that infinitely many sheets of the same branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056023.png" /> are joined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056024.png" />; this branch is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056025.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056026.png" /> by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056027.png" />.
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A special form of a [[Branch point|branch point]] $  a $
 +
of an analytic function $  f ( z) $
 +
of one complex variable $  z $,  
 +
when for no finite number of successive circuits in the same direction about $  a $
 +
the [[Analytic continuation|analytic continuation]] of some element of $  f ( z) $
 +
returns to the original element. More precisely, an isolated singular point $  a $
 +
is called a logarithmic branch point for $  f ( z) $
 +
if there exist: 1) an annulus $  V = \{ {z } : {0 < | z - a | < \rho } \} $
 +
in which $  f ( z) $
 +
can be analytically continued along any path; and 2) a point $  z _ {1} \in V $
 +
and an element of $  f ( z) $
 +
in the form of a power series $  \Pi ( z _ {1} ;  r ) = \sum _ {\nu = 0 }  ^  \infty  c _  \nu  ( z - z _ {1} )  ^  \nu  $
 +
with centre $  z _ {1} $
 +
and radius of convergence $  r > 0 $,  
 +
the analytic continuation of which along the circle $  | z - a | = | z _ {1} - a | $,  
 +
taken arbitrarily many times in the same direction, never returns to the original element $  \Pi ( z _ {1} ;  r ) $.  
 +
In the case of a logarithmic branch point at infinity, $  a = \infty $,  
 +
instead of $  V $
 +
one must consider a neighbourhood $  V  ^  \prime  = \{ {z } : {| z | > \rho } \} $.  
 +
Logarithmic branch points belong to the class of transcendental branch points (cf. [[Transcendental branch point|Transcendental branch point]]). The behaviour of the Riemann surface $  R $
 +
of a function $  f ( z) $
 +
in the presence of a logarithmic branch point $  a $
 +
is characterized by the fact that infinitely many sheets of the same branch of $  R $
 +
are joined over $  a $;  
 +
this branch is defined in $  V $
 +
or $  V  ^  \prime  $
 +
by the elements $  \Pi ( z _ {1} ;  r ) $.
  
 
See also [[Singular point|Singular point]] of an analytic function.
 
See also [[Singular point|Singular point]] of an analytic function.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  pp. Chapt. 8  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  pp. Chapt. 8  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056028.png" /> has a logarithmic branch point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056030.png" /> is the (multiple-valued) [[Logarithmic function|logarithmic function]] of a complex variable.
+
The function $  \mathop{\rm Ln} ( z - z _ {0} ) $
 +
has a logarithmic branch point at $  z _ {0} $,  
 +
where $  \mathop{\rm Ln} $
 +
is the (multiple-valued) [[Logarithmic function|logarithmic function]] of a complex variable.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. Chapt. 8</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. Chapt. 8</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


branch point of infinite order

A special form of a branch point $ a $ of an analytic function $ f ( z) $ of one complex variable $ z $, when for no finite number of successive circuits in the same direction about $ a $ the analytic continuation of some element of $ f ( z) $ returns to the original element. More precisely, an isolated singular point $ a $ is called a logarithmic branch point for $ f ( z) $ if there exist: 1) an annulus $ V = \{ {z } : {0 < | z - a | < \rho } \} $ in which $ f ( z) $ can be analytically continued along any path; and 2) a point $ z _ {1} \in V $ and an element of $ f ( z) $ in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $ with centre $ z _ {1} $ and radius of convergence $ r > 0 $, the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, taken arbitrarily many times in the same direction, never returns to the original element $ \Pi ( z _ {1} ; r ) $. In the case of a logarithmic branch point at infinity, $ a = \infty $, instead of $ V $ one must consider a neighbourhood $ V ^ \prime = \{ {z } : {| z | > \rho } \} $. Logarithmic branch points belong to the class of transcendental branch points (cf. Transcendental branch point). The behaviour of the Riemann surface $ R $ of a function $ f ( z) $ in the presence of a logarithmic branch point $ a $ is characterized by the fact that infinitely many sheets of the same branch of $ R $ are joined over $ a $; this branch is defined in $ V $ or $ V ^ \prime $ by the elements $ \Pi ( z _ {1} ; r ) $.

See also Singular point of an analytic function.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)

Comments

The function $ \mathop{\rm Ln} ( z - z _ {0} ) $ has a logarithmic branch point at $ z _ {0} $, where $ \mathop{\rm Ln} $ is the (multiple-valued) logarithmic function of a complex variable.

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8
How to Cite This Entry:
Logarithmic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_branch_point&oldid=47700
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article