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Difference between revisions of "Algebraic logarithmic singular point"

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An isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115801.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115802.png" /> such that in a neighbourhood of it the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115803.png" /> may be represented as the sum of a finite number of terms of the form
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<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/a011580/a0115804.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115805.png" /> is a complex number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115806.png" /> is a non-negative integer, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115807.png" /> is a regular analytic function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115808.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011580/a0115809.png" />.
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An isolated singular point  $  z _ {0} $
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of an analytic function  $  f(z) $
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such that in a neighbourhood of it the function  $  f(z) $
 +
may be represented as the sum of a finite number of terms of the form
 +
 
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$$
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( z - z _ {0} )  ^ {-s} [  \mathop{\rm ln} ( z - z _ {0} ) ]  ^ {k}
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g (z) ,
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$$
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where  $  s $
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is a complex number, $  k $
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is a non-negative integer, and $  g(z) $
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is a regular analytic function at the point $  z _ {0} $
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with $  g ( z _ {0} ) \neq 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR></table>

Latest revision as of 16:10, 1 April 2020


An isolated singular point $ z _ {0} $ of an analytic function $ f(z) $ such that in a neighbourhood of it the function $ f(z) $ may be represented as the sum of a finite number of terms of the form

$$ ( z - z _ {0} ) ^ {-s} [ \mathop{\rm ln} ( z - z _ {0} ) ] ^ {k} g (z) , $$

where $ s $ is a complex number, $ k $ is a non-negative integer, and $ g(z) $ is a regular analytic function at the point $ z _ {0} $ with $ g ( z _ {0} ) \neq 0 $.

References

[1] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3
How to Cite This Entry:
Algebraic logarithmic singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_logarithmic_singular_point&oldid=45063
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article