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Difference between revisions of "Non-singular boundary point"

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''regular boundary point''
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An accessible boundary point (cf. [[Attainable boundary point|Attainable boundary point]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672901.png" /> of the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672902.png" /> of a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672903.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672904.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672905.png" /> has an [[Analytic continuation|analytic continuation]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672906.png" /> along any path inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672907.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672908.png" />. In other words, a non-singular boundary point is accessible, but not singular. See also [[Singular point|Singular point]] of an analytic function.
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''regular boundary point''
  
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An accessible boundary point (cf. [[Attainable boundary point|Attainable boundary point]])  $  \zeta $
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of the domain of definition  $  D $
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of a single-valued analytic function  $  f ( z) $
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of a complex variable  $  z $
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such that  $  f ( z ) $
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has an [[Analytic continuation|analytic continuation]] to  $  \zeta $
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along any path inside  $  D $
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to  $  \zeta $.
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In other words, a non-singular boundary point is accessible, but not singular. See also [[Singular point|Singular point]] of an analytic function.
  
 
====Comments====
 
====Comments====
Note that the same point in the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n0672909.png" /> may give rise to several different accessible boundary points, some of which may be singular, others regular. E.g., consider the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729010.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729012.png" /> is the principal value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729013.png" />. Then  "above"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729014.png" /> there are two accessible boundary points: one singular, corresponding to approach along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729016.png" />; one regular, corresponding to approach along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067290/n06729018.png" />.
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Note that the same point in the boundary of $  D $
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may give rise to several different accessible boundary points, some of which may be singular, others regular. E.g., consider the domain $  D = \mathbf C \setminus  ( - \infty , 0 ] $,  
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and the function $  f ( z) = ( h ( z) - \pi i )  ^ {-} 1 $,  
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where $  h $
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is the principal value of $  \mathop{\rm log}  z $.  
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Then  "above"   $ - 1 $
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there are two accessible boundary points: one singular, corresponding to approach along $  z = - 1 + i t $,  
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0 \leq  t \leq  1 $;  
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one regular, corresponding to approach along $  z = - 1 - i t $,  
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0 \leq  t \leq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapts. 2; 8  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapts. 2; 8  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


regular boundary point

An accessible boundary point (cf. Attainable boundary point) $ \zeta $ of the domain of definition $ D $ of a single-valued analytic function $ f ( z) $ of a complex variable $ z $ such that $ f ( z ) $ has an analytic continuation to $ \zeta $ along any path inside $ D $ to $ \zeta $. In other words, a non-singular boundary point is accessible, but not singular. See also Singular point of an analytic function.

Comments

Note that the same point in the boundary of $ D $ may give rise to several different accessible boundary points, some of which may be singular, others regular. E.g., consider the domain $ D = \mathbf C \setminus ( - \infty , 0 ] $, and the function $ f ( z) = ( h ( z) - \pi i ) ^ {-} 1 $, where $ h $ is the principal value of $ \mathop{\rm log} z $. Then "above" $ - 1 $ there are two accessible boundary points: one singular, corresponding to approach along $ z = - 1 + i t $, $ 0 \leq t \leq 1 $; one regular, corresponding to approach along $ z = - 1 - i t $, $ 0 \leq t \leq 1 $.

References

[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapts. 2; 8 (Translated from Russian)
How to Cite This Entry:
Non-singular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-singular_boundary_point&oldid=48003
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article