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''singular point of multi-valued character''
 
''singular point of multi-valued character''
  
An [[Isolated singular point|isolated singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175001.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175002.png" /> of one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175003.png" /> such that the [[Analytic continuation|analytic continuation]] of an arbitrary function element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175004.png" /> along a closed path which encircles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175005.png" /> yields new elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175006.png" />. More exactly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175007.png" /> is said to be a branch point if there exist: 1) an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175008.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b0175009.png" /> can be analytically extended along any path; 2) a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750010.png" /> and some function element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750011.png" /> represented by a power series
+
An [[Isolated singular point|isolated singular point]] $  a $
 +
of an analytic function $  f(z) $
 +
of one complex variable $  z $
 +
such that the [[Analytic continuation|analytic continuation]] of an arbitrary function element of $  f(z) $
 +
along a closed path which encircles $  a $
 +
yields new elements of $  f(z) $.  
 +
More exactly, $  a $
 +
is said to be a branch point if there exist: 1) an annulus $  V= \{ {z } : {0 < | z - a | < \rho } \} $
 +
in which $  f(z) $
 +
can be analytically extended along any path; 2) a point $  z _ {1} \in V $
 +
and some function element of $  f(z) $
 +
represented by a 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/b/b017/b017500/b01750012.png" /></td> </tr></table>
+
$$
 +
\Pi (z _ {1} ; r)  = \
 +
\sum _ {v = 0 } ^  \infty 
 +
c _ {v} (z - z _ {1} )  ^ {v}
 +
$$
  
with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750013.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750014.png" />, the analytic continuation of which along the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750015.png" />, going around the path once in, say, the positive direction, yields a new element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750016.png" /> different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750017.png" />. If, after a minimum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750018.png" /> of such rounds the initial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750019.png" /> is again obtained, this is also true of all elements of the branch (cf. [[Branch of an analytic function|Branch of an analytic function]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750020.png" /> defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750021.png" /> by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750022.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750023.png" /> is a branch point of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750024.png" /> of this branch. In a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750025.png" /> of a branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750026.png" /> of finite order this branch is represented by a generalized Laurent series, or Puiseux series:
+
with centre $  z _ {1} $
 +
and radius of convergence $  r > 0 $,  
 +
the analytic continuation of which along the circle $  | z - a | = | z _ {1} - a | $,  
 +
going around the path once in, say, the positive direction, yields a new element $  \Pi ^ { \prime } (z _ {1} ;  r  ^  \prime  ) $
 +
different from $  \Pi (z _ {1} ;  r) $.  
 +
If, after a minimum number $  k > 1 $
 +
of such rounds the initial element $  \Pi (z _ {1} ;  r) $
 +
is again obtained, this is also true of all elements of the branch (cf. [[Branch of an analytic function|Branch of an analytic function]]) of $  f(z) $
 +
defined in $  V $
 +
by the element $  \Pi (z _ {1} ;  r) $.  
 +
In such a case $  a $
 +
is a branch point of finite order $  k - 1 $
 +
of this branch. In a punctured neighbourhood $  V $
 +
of a branch point $  a $
 +
of finite order this branch is represented by a generalized Laurent series, or 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/b/b017/b017500/b01750027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f (z)  = \
 +
\sum _ {v = - \infty } ^ { {+ }  \infty }
 +
b _ {v} (z - a) ^ {v/k} ,
 +
\  z \in V.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750028.png" /> is an improper branch point of a finite order, then the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750029.png" /> is representable in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750030.png" /> by an analogue of the series (1):
+
If $  a = \infty $
 +
is an improper branch point of a finite order, then the branch of $  f(z) $
 +
is representable in some neighbourhood $  V ^ { \prime } = \{ {z } : {| z | > \rho } \} $
 +
by an analogue of the series (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/b/b017/b017500/b01750031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f (z)  = \
 +
\sum _ {v = - \infty } ^ { {+ }  \infty }
 +
b _ {v} z  ^ {-v/k} ,
 +
\  z \in V  ^  \prime  .
 +
$$
  
The behaviour of the [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750033.png" /> over a branch point of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750034.png" /> is characterized by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750035.png" /> sheets of the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750036.png" /> defined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750037.png" /> come together over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750038.png" />. At the same time the behaviour of other branches of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750040.png" /> may be altogether different.
+
The behaviour of the [[Riemann surface|Riemann surface]] $  R $
 +
of $  f(z) $
 +
over a branch point of finite order $  a $
 +
is characterized by the fact that $  k $
 +
sheets of the branch of $  f(z) $
 +
defined by the element $  \Pi (z _ {1} ;  r) $
 +
come together over $  a $.  
 +
At the same time the behaviour of other branches of $  R $
 +
over $  a $
 +
may be altogether different.
  
If the series (1) or (2) contains only a finite number of non-zero coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750041.png" /> with negative indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750043.png" /> is an [[Algebraic branch point|algebraic branch point]] or an algebraic singular point. Such a branch point of finite order is also characterized by the fact that as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750044.png" /> in whatever manner, the values of all elements of the branch defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750046.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750047.png" /> tend to a definite finite or infinite limit.
+
If the series (1) or (2) contains only a finite number of non-zero coefficients b _ {v} $
 +
with negative indices $  v $,  
 +
$  a $
 +
is an [[Algebraic branch point|algebraic branch point]] or an algebraic singular point. Such a branch point of finite order is also characterized by the fact that as $  z \rightarrow a $
 +
in whatever manner, the values of all elements of the branch defined by $  \Pi (z _ {1} ;  r) $
 +
in $  V $
 +
or $  V ^ { \prime } $
 +
tend to a definite finite or infinite limit.
  
Example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750049.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750050.png" />.
+
Example: $  f (z) = z  ^ {1/k} $,  
 +
where $  k > 1 $
 +
is a natural number, $  a = 0, \infty $.
  
If the series (1) or (2) contain an infinite number of non-zero coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750051.png" /> with negative indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750052.png" />, the branch points of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750053.png" /> belong the class of transcendental branch points.
+
If the series (1) or (2) contain an infinite number of non-zero coefficients b _ {v} $
 +
with negative indices $  v $,  
 +
the branch points of finite order $  a $
 +
belong the class of transcendental branch points.
  
Example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750055.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750056.png" />.
+
Example: $  f (z) = \mathop{\rm exp} (1/z)  ^ {1/k} $,  
 +
where $  k > 1 $
 +
is a natural number, $  a = 0 $.
  
Finally, if it is impossible to return to the initial element after a finite number of turns, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750057.png" /> is said to be a [[Logarithmic branch point|logarithmic branch point]] or a branch point of infinite order, and is also a transcendental branch point.
+
Finally, if it is impossible to return to the initial element after a finite number of turns, $  a $
 +
is said to be a [[Logarithmic branch point|logarithmic branch point]] or a branch point of infinite order, and is also a transcendental branch point.
  
Example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750058.png" />.
+
Example: $  f(z) = \mathop{\rm Ln}  z, a = 0, \infty $.
  
Infinitely many sheets of the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750059.png" /> defined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750060.png" /> come together over a logarithmic branch point.
+
Infinitely many sheets of the branch of $  f(z) $
 +
defined by the element $  \Pi (z _ {1} ;  r) $
 +
come together over a logarithmic branch point.
  
In the case of an analytic function of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750063.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750064.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750065.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750066.png" /> is said to be a branch point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750069.png" />, if it is a branch point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750070.png" /> of the, generally many-sheeted, [[Domain of holomorphy|domain of holomorphy]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750071.png" />. Unlike in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750072.png" />, branch points, just like other singular points of analytic functions (cf. [[Singular point|Singular point]]), cannot be isolated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017500/b01750073.png" />.
+
In the case of an analytic function of several complex variables $  f(z) $,  
 +
$  z = (z _ {1} \dots z _ {n} ) $,  
 +
$  n \geq  2 $,  
 +
a point $  a $
 +
of the space $  \mathbf C  ^ {n} $
 +
or $  \mathbf C P  ^ {n} $
 +
is said to be a branch point of order $  m $,  
 +
$  1 \leq  m \leq  \infty $,  
 +
if it is a branch point of order $  m $
 +
of the, generally many-sheeted, [[Domain of holomorphy|domain of holomorphy]] of $  f(z) $.  
 +
Unlike in the case $  n=1 $,  
 +
branch points, just like other singular points of analytic functions (cf. [[Singular point|Singular point]]), cannot be isolated if $  n \geq  2 $.
  
 
====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><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Fuks,  "Theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1963)  (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><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Fuks,  "Theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>

Revision as of 06:29, 30 May 2020


singular point of multi-valued character

An isolated singular point $ a $ of an analytic function $ f(z) $ of one complex variable $ z $ such that the analytic continuation of an arbitrary function element of $ f(z) $ along a closed path which encircles $ a $ yields new elements of $ f(z) $. More exactly, $ a $ is said to be a branch point if there exist: 1) an annulus $ V= \{ {z } : {0 < | z - a | < \rho } \} $ in which $ f(z) $ can be analytically extended along any path; 2) a point $ z _ {1} \in V $ and some function element of $ f(z) $ represented by a power series

$$ \Pi (z _ {1} ; r) = \ \sum _ {v = 0 } ^ \infty c _ {v} (z - z _ {1} ) ^ {v} $$

with centre $ z _ {1} $ and radius of convergence $ r > 0 $, the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, going around the path once in, say, the positive direction, yields a new element $ \Pi ^ { \prime } (z _ {1} ; r ^ \prime ) $ different from $ \Pi (z _ {1} ; r) $. If, after a minimum number $ k > 1 $ of such rounds the initial element $ \Pi (z _ {1} ; r) $ is again obtained, this is also true of all elements of the branch (cf. Branch of an analytic function) of $ f(z) $ defined in $ V $ by the element $ \Pi (z _ {1} ; r) $. In such a case $ a $ is a branch point of finite order $ k - 1 $ of this branch. In a punctured neighbourhood $ V $ of a branch point $ a $ of finite order this branch is represented by a generalized Laurent series, or Puiseux series:

$$ \tag{1 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} (z - a) ^ {v/k} , \ z \in V. $$

If $ a = \infty $ is an improper branch point of a finite order, then the branch of $ f(z) $ is representable in some neighbourhood $ V ^ { \prime } = \{ {z } : {| z | > \rho } \} $ by an analogue of the series (1):

$$ \tag{2 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} z ^ {-v/k} , \ z \in V ^ \prime . $$

The behaviour of the Riemann surface $ R $ of $ f(z) $ over a branch point of finite order $ a $ is characterized by the fact that $ k $ sheets of the branch of $ f(z) $ defined by the element $ \Pi (z _ {1} ; r) $ come together over $ a $. At the same time the behaviour of other branches of $ R $ over $ a $ may be altogether different.

If the series (1) or (2) contains only a finite number of non-zero coefficients $ b _ {v} $ with negative indices $ v $, $ a $ is an algebraic branch point or an algebraic singular point. Such a branch point of finite order is also characterized by the fact that as $ z \rightarrow a $ in whatever manner, the values of all elements of the branch defined by $ \Pi (z _ {1} ; r) $ in $ V $ or $ V ^ { \prime } $ tend to a definite finite or infinite limit.

Example: $ f (z) = z ^ {1/k} $, where $ k > 1 $ is a natural number, $ a = 0, \infty $.

If the series (1) or (2) contain an infinite number of non-zero coefficients $ b _ {v} $ with negative indices $ v $, the branch points of finite order $ a $ belong the class of transcendental branch points.

Example: $ f (z) = \mathop{\rm exp} (1/z) ^ {1/k} $, where $ k > 1 $ is a natural number, $ a = 0 $.

Finally, if it is impossible to return to the initial element after a finite number of turns, $ a $ is said to be a logarithmic branch point or a branch point of infinite order, and is also a transcendental branch point.

Example: $ f(z) = \mathop{\rm Ln} z, a = 0, \infty $.

Infinitely many sheets of the branch of $ f(z) $ defined by the element $ \Pi (z _ {1} ; r) $ come together over a logarithmic branch point.

In the case of an analytic function of several complex variables $ f(z) $, $ z = (z _ {1} \dots z _ {n} ) $, $ n \geq 2 $, a point $ a $ of the space $ \mathbf C ^ {n} $ or $ \mathbf C P ^ {n} $ is said to be a branch point of order $ m $, $ 1 \leq m \leq \infty $, if it is a branch point of order $ m $ of the, generally many-sheeted, domain of holomorphy of $ f(z) $. Unlike in the case $ n=1 $, branch points, just like other singular points of analytic functions (cf. Singular point), cannot be isolated if $ n \geq 2 $.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)
[2] B.A. Fuks, "Theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian)
How to Cite This Entry:
Branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branch_point&oldid=16924
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article