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''for an element of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527801.png" />''
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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527802.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527803.png" />-plane satisfying the following properties: 1) the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527804.png" /> does not have an [[Analytic continuation|analytic continuation]] along any path to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527805.png" />; and 2) there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527806.png" /> such that analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527807.png" /> is possible along any path in the punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527808.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i0527809.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
If a new element is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278010.png" /> is continued analytically along a closed path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278011.png" /> encircling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278012.png" />, for example along the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278015.png" /> is called a [[Branch point|branch point]], or an isolated singular point of multi-valued character. Otherwise the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278016.png" /> defines a single-valued analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278018.png" /> is called an isolated singular point of single-valued character. In a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278019.png" /> of an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278020.png" /> of single-valued character, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278021.png" /> can be expanded in a [[Laurent series|Laurent series]]:
+
''for an element of an analytic function $  f ( z) $''
  
<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/i052780/i05278022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A point  $  a $
 +
in the complex  $  z $-
 +
plane satisfying the following properties: 1) the element of  $  f ( z) $
 +
does not have an [[Analytic continuation|analytic continuation]] along any path to  $  a $;  
 +
and 2) there exists a number  $  R > 0 $
 +
such that analytic continuation of  $  f ( z) $
 +
is possible along any path in the punctured neighbourhood  $  U = \{ {z \in \mathbf C } : {0 < | z - a | < R } \} $
 +
of  $  a $.
  
with regular part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278023.png" /> and principal part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278024.png" />. The behaviour of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278025.png" /> in a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278026.png" /> of an isolated singular point of single-valued character is determined, in principle, by the principal part of its Laurent series. If all the coefficients of the principal part are zero, then on setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278027.png" /> one gets a single-valued analytic function in a full neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278028.png" />. This case of practical absence of a singularity is also characterized by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278029.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278030.png" />, or by the fact that the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278032.png" />, exists and is finite.
+
If a new element is obtained when  $  f ( z) $
 +
is continued analytically along a closed path in  $  U $
 +
encircling  $  a $,
 +
for example along the circle  $  | z - a | = \rho $,
 +
0 < \rho < R $,
 +
then  $  a $
 +
is called a [[Branch point|branch point]], or an isolated singular point of multi-valued character. Otherwise the element of $  f ( z) $
 +
defines a single-valued analytic function in $  U $
 +
and  $  a $
 +
is called an isolated singular point of single-valued character. In a punctured neighbourhood  $  U $
 +
of an isolated singular point  $  a $
 +
of single-valued character,  $  f ( z) $
 +
can be expanded in a [[Laurent series|Laurent series]]:
  
If among the coefficients of the principal part only finitely many are non-zero, and that with smallest index is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278034.png" /> is a pole of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278035.png" /> (cf. [[Pole (of a function)|Pole (of a function)]]). A pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278036.png" /> is also characterized by the fact that
+
$$ \tag{1 }
 +
f ( z)  = \
 +
\sum _ {k = - \infty } ^ { {+ }  \infty }
 +
c _ {k} ( z - a) ^ {k}
 +
$$
  
<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/i052780/i05278037.png" /></td> </tr></table>
+
with regular part  $  f _ {1} ( z) = \sum _ {k = 0 }  ^ {+ \infty } c _ {k} ( z - a)  ^ {k} $
 +
and principal part  $  f _ {2} ( z) = \sum _ {k = - \infty }  ^ {-} 1 c _ {k} ( z- a)  ^ {k} $.
 +
The behaviour of an analytic function  $  f ( z) $
 +
in a punctured neighbourhood  $  U $
 +
of an isolated singular point of single-valued character is determined, in principle, by the principal part of its Laurent series. If all the coefficients of the principal part are zero, then on setting  $  f ( a) = c _ {0} $
 +
one gets a single-valued analytic function in a full neighbourhood of  $  a $.
 +
This case of practical absence of a singularity is also characterized by the fact that  $  f ( z) $
 +
is bounded in  $  U $,
 +
or by the fact that the limit  $  \lim\limits _ {z \rightarrow a }  f ( z) = c _ {0} $,
 +
$  z \in U $,
 +
exists and is finite.
  
Finally, if there are infinitely many non-zero coefficients in the principal part, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278038.png" /> is an [[Essential singular point|essential singular point]]. In this case the following limit does not exist, neither finite nor infinite:
+
If among the coefficients of the principal part only finitely many are non-zero, and that with smallest index is  $  c _ {-} m \neq 0 $,
 +
then $  a $
 +
is a pole of order  $  m $(
 +
cf. [[Pole (of a function)|Pole (of a function)]]). A pole  $  a $
 +
is also characterized by the fact that
  
<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/i052780/i05278039.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {z \rightarrow a }  f ( z)  = \infty ,\ \
 +
z \in U.
 +
$$
  
For an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278040.png" /> at infinity of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278041.png" />, a punctured neighbourhood has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278042.png" />, and the Laurent series is
+
Finally, if there are infinitely many non-zero coefficients in the principal part, then  $  a $
 +
is an [[Essential singular point|essential singular point]]. In this case the following limit does not exist, neither finite nor infinite:
  
<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/i052780/i05278043.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {z \rightarrow a }  f ( z),\  z \in U.
 +
$$
  
Here the regular part is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278044.png" /> and the principal part is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278045.png" />. With these conditions, the above descriptions of the classification and criteria for the type of an isolated singular point carry over to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278046.png" /> without further change (see also [[Residue of an analytic function|Residue of an analytic function]]). It should be noted that the elements of different branches of the [[Complete analytic function|complete analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278047.png" /> at one and the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278048.png" /> may have singularities of completely-different types.
+
For an isolated singular point $  a = \infty $
 +
at infinity of the element  $  f( z) $,
 +
a punctured neighbourhood has the form  $  U = \{ {z \in \mathbf C } : {r < | z | < \infty } \} $,
 +
and the Laurent series is
  
Holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278049.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278051.png" />, cannot have isolated singular points. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052780/i05278052.png" />, the singular points form infinite sets of singularities.
+
$$
 +
f ( z)  = \
 +
\sum _ {k = - \infty } ^ { {+ }  \infty }
 +
c _ {k} z  ^ {k} .
 +
$$
  
====References====
+
Here the regular part is  $  f _ {1} = \sum _ {k = - \infty }  ^ {0} c _ {k} z  ^ {k} $
<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"> B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow (1976(In Russian)</TD></TR></table>
+
and the principal part is  $  f _ {2} ( z) = \sum _ {k = 1 }  ^ {+ \infty } c _ {k} z ^ {k} $.  
 +
With these conditions, the above descriptions of the classification and criteria for the type of an isolated singular point carry over to the case $ a = \infty $
 +
without further change (see also [[Residue of an analytic function|Residue of an analytic function]]). It should be noted that the elements of different branches of the [[Complete analytic function|complete analytic function]$ f ( z) $
 +
at one and the same point $  a \in \mathbf C $
 +
may have singularities of completely-different types.
  
 +
Holomorphic functions  $  f ( z) $
 +
of several complex variables  $  z = ( z _ {1} \dots z _ {n} ) $,
 +
$  n \geq  2 $,
 +
cannot have isolated singular points. For  $  n \geq  2 $,
 +
the singular points form infinite sets of singularities.
  
 +
====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) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Ahlfors,   "Complex analysis" , McGraw-Hill (1979) pp. 241</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{MR|0510197}} {{MR|1535085}} {{MR|0188405}} {{MR|1570643}} {{MR|1528598}} {{MR|0054016}} {{ZBL|0395.30001}} </TD></TR></table>

Latest revision as of 22:13, 5 June 2020


for an element of an analytic function $ f ( z) $

A point $ a $ in the complex $ z $- plane satisfying the following properties: 1) the element of $ f ( z) $ does not have an analytic continuation along any path to $ a $; and 2) there exists a number $ R > 0 $ such that analytic continuation of $ f ( z) $ is possible along any path in the punctured neighbourhood $ U = \{ {z \in \mathbf C } : {0 < | z - a | < R } \} $ of $ a $.

If a new element is obtained when $ f ( z) $ is continued analytically along a closed path in $ U $ encircling $ a $, for example along the circle $ | z - a | = \rho $, $ 0 < \rho < R $, then $ a $ is called a branch point, or an isolated singular point of multi-valued character. Otherwise the element of $ f ( z) $ defines a single-valued analytic function in $ U $ and $ a $ is called an isolated singular point of single-valued character. In a punctured neighbourhood $ U $ of an isolated singular point $ a $ of single-valued character, $ f ( z) $ can be expanded in a Laurent series:

$$ \tag{1 } f ( z) = \ \sum _ {k = - \infty } ^ { {+ } \infty } c _ {k} ( z - a) ^ {k} $$

with regular part $ f _ {1} ( z) = \sum _ {k = 0 } ^ {+ \infty } c _ {k} ( z - a) ^ {k} $ and principal part $ f _ {2} ( z) = \sum _ {k = - \infty } ^ {-} 1 c _ {k} ( z- a) ^ {k} $. The behaviour of an analytic function $ f ( z) $ in a punctured neighbourhood $ U $ of an isolated singular point of single-valued character is determined, in principle, by the principal part of its Laurent series. If all the coefficients of the principal part are zero, then on setting $ f ( a) = c _ {0} $ one gets a single-valued analytic function in a full neighbourhood of $ a $. This case of practical absence of a singularity is also characterized by the fact that $ f ( z) $ is bounded in $ U $, or by the fact that the limit $ \lim\limits _ {z \rightarrow a } f ( z) = c _ {0} $, $ z \in U $, exists and is finite.

If among the coefficients of the principal part only finitely many are non-zero, and that with smallest index is $ c _ {-} m \neq 0 $, then $ a $ is a pole of order $ m $( cf. Pole (of a function)). A pole $ a $ is also characterized by the fact that

$$ \lim\limits _ {z \rightarrow a } f ( z) = \infty ,\ \ z \in U. $$

Finally, if there are infinitely many non-zero coefficients in the principal part, then $ a $ is an essential singular point. In this case the following limit does not exist, neither finite nor infinite:

$$ \lim\limits _ {z \rightarrow a } f ( z),\ z \in U. $$

For an isolated singular point $ a = \infty $ at infinity of the element $ f( z) $, a punctured neighbourhood has the form $ U = \{ {z \in \mathbf C } : {r < | z | < \infty } \} $, and the Laurent series is

$$ f ( z) = \ \sum _ {k = - \infty } ^ { {+ } \infty } c _ {k} z ^ {k} . $$

Here the regular part is $ f _ {1} = \sum _ {k = - \infty } ^ {0} c _ {k} z ^ {k} $ and the principal part is $ f _ {2} ( z) = \sum _ {k = 1 } ^ {+ \infty } c _ {k} z ^ {k} $. With these conditions, the above descriptions of the classification and criteria for the type of an isolated singular point carry over to the case $ a = \infty $ without further change (see also Residue of an analytic function). It should be noted that the elements of different branches of the complete analytic function $ f ( z) $ at one and the same point $ a \in \mathbf C $ may have singularities of completely-different types.

Holomorphic functions $ f ( z) $ of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 2 $, cannot have isolated singular points. For $ n \geq 2 $, the singular points form infinite sets of singularities.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001

Comments

References

[a1] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01
[a2] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 MR0510197 MR1535085 MR0188405 MR1570643 MR1528598 MR0054016 Zbl 0395.30001
How to Cite This Entry:
Isolated singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_singular_point&oldid=12648
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article