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Difference between revisions of "Transcendental branch point"

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''of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936001.png" />''
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$#C+1 = 11 : ~/encyclopedia/old_files/data/T093/T.0903600 Transcendental branch point
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A [[Branch point|branch point]] that is not an [[Algebraic branch point|algebraic branch point]]. In other words, it is either a branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936002.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936003.png" /> at which, however, there does not exist a finite or infinite limit
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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/t/t093/t093600/t0936004.png" /></td> </tr></table>
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''of an analytic function  $  f ( z) $''
  
or a [[Logarithmic branch point|logarithmic branch point]] of infinite order. For example, the first possibility is realized at the transcendental branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936005.png" /> for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936006.png" />, the second for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936007.png" />.
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A [[Branch point|branch point]] that is not an [[Algebraic branch point|algebraic branch point]]. In other words, it is either a branch point $  a $
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of finite order  $  k > 0 $
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at which, however, there does not exist a finite or infinite limit
  
In the first case the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936008.png" /> can be expanded in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936009.png" /> in the form of a [[Puiseux series|Puiseux series]]
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$$
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\lim\limits _ {\begin{array}{c}
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z \rightarrow a \\
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z \neq a  
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\end{array}
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}  f ( z),
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$$
  
<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/t/t093/t093600/t09360010.png" /></td> </tr></table>
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or a [[Logarithmic branch point|logarithmic branch point]] of infinite order. For example, the first possibility is realized at the transcendental branch point  $  a = 0 $
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for the function  $  \mathop{\rm exp} ( 1/z  ^ {1/k} ) $,
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the second for the function  $  \mathop{\rm ln}  z $.
  
with an infinite number of non-zero coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t09360011.png" /> with negative indices.
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In the first case the function  $  f ( z) $
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can be expanded in a neighbourhood of  $  a $
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in the form of a [[Puiseux series|Puiseux series]]
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$$
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f ( z)  =  \sum _ {n = - \infty } ^ { {+ }  \infty } c _ {n} ( z - a)  ^ {n/k}
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$$
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with an infinite number of non-zero coefficients $  c _ {n} $
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with negative indices.
  
 
====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)  (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)  (Translated from Russian)</TD></TR></table>

Revision as of 08:26, 6 June 2020


of an analytic function $ f ( z) $

A branch point that is not an algebraic branch point. In other words, it is either a branch point $ a $ of finite order $ k > 0 $ at which, however, there does not exist a finite or infinite limit

$$ \lim\limits _ {\begin{array}{c} z \rightarrow a \\ z \neq a \end{array} } f ( z), $$

or a logarithmic branch point of infinite order. For example, the first possibility is realized at the transcendental branch point $ a = 0 $ for the function $ \mathop{\rm exp} ( 1/z ^ {1/k} ) $, the second for the function $ \mathop{\rm ln} z $.

In the first case the function $ f ( z) $ can be expanded in a neighbourhood of $ a $ in the form of a Puiseux series

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

with an infinite number of non-zero coefficients $ c _ {n} $ with negative indices.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Transcendental branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_branch_point&oldid=49007
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article