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''boundary value along a non-tangential path''
 
''boundary value along a non-tangential path''
  
The value associated to a complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125101.png" /> defined in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125102.png" /> at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125103.png" />, equal to the limit
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The value associated to a complex function $  f (x) $
 +
defined in the unit disc $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
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at a boundary point $  \zeta = e ^ {i \theta } $,  
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equal to the limit
  
<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/a012/a012510/a0125104.png" /></td> </tr></table>
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$$
 +
\lim\limits _ {
 +
\begin{array}{c}
 +
z \in S \\
 +
z \rightarrow \zeta
 +
\end{array}
 +
} \
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f (z)  = f  ^ {*} ( \zeta )
 +
$$
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125105.png" /> on the set of points of the angular domain
+
of $  f (z) $
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on the set of points of the angular domain
  
<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/a012/a012510/a0125106.png" /></td> </tr></table>
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$$
 +
S ( \zeta , \epsilon )  = \
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\left \{ {z = r e ^ {i \phi } \in D } : {|
 +
\mathop{\rm arg} ( e ^ {i \theta } - z ) | <  
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\frac \pi {2}
  
under the condition that this limit exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125108.png" />, and hence does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a0125109.png" />. The term is sometimes applied in a more general sense to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251010.png" /> given in an arbitrary (including a higher-dimensional) domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251011.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251012.png" /> one takes the intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251013.png" /> of an angular (or conical) domain with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251014.png" />, with axis normal to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251016.png" /> and with angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012510/a01251018.png" />.
+
- \epsilon } \right \}
 +
$$
 +
 
 +
under the condition that this limit exists for all $  \epsilon $,
 +
$  0 < \epsilon < \pi / 2 $,  
 +
and hence does not depend on $  \epsilon $.  
 +
The term is sometimes applied in a more general sense to functions $  f (z) $
 +
given in an arbitrary (including a higher-dimensional) domain $  D $;  
 +
for $  S ( \zeta , \epsilon ) $
 +
one takes the intersection with $  D $
 +
of an angular (or conical) domain with vertex $  \zeta \in \partial  D $,  
 +
with axis normal to the boundary $  \partial  D $
 +
at $  \zeta $
 +
and with angle $  \pi / 2 - \epsilon $,  
 +
$  0 < \epsilon < \pi / 2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (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" , '''1–2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
An angular boundary value is also called a non-tangential boundary value. Cf. [[Boundary properties of analytic functions|Boundary properties of analytic functions]].
 
An angular boundary value is also called a non-tangential boundary value. Cf. [[Boundary properties of analytic functions|Boundary properties of analytic functions]].

Latest revision as of 18:47, 5 April 2020


boundary value along a non-tangential path

The value associated to a complex function $ f (x) $ defined in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ at a boundary point $ \zeta = e ^ {i \theta } $, equal to the limit

$$ \lim\limits _ { \begin{array}{c} z \in S \\ z \rightarrow \zeta \end{array} } \ f (z) = f ^ {*} ( \zeta ) $$

of $ f (z) $ on the set of points of the angular domain

$$ S ( \zeta , \epsilon ) = \ \left \{ {z = r e ^ {i \phi } \in D } : {| \mathop{\rm arg} ( e ^ {i \theta } - z ) | < \frac \pi {2} - \epsilon } \right \} $$

under the condition that this limit exists for all $ \epsilon $, $ 0 < \epsilon < \pi / 2 $, and hence does not depend on $ \epsilon $. The term is sometimes applied in a more general sense to functions $ f (z) $ given in an arbitrary (including a higher-dimensional) domain $ D $; for $ S ( \zeta , \epsilon ) $ one takes the intersection with $ D $ of an angular (or conical) domain with vertex $ \zeta \in \partial D $, with axis normal to the boundary $ \partial D $ at $ \zeta $ and with angle $ \pi / 2 - \epsilon $, $ 0 < \epsilon < \pi / 2 $.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian)
[2] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

Comments

An angular boundary value is also called a non-tangential boundary value. Cf. Boundary properties of analytic functions.

How to Cite This Entry:
Angular boundary value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Angular_boundary_value&oldid=45187
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article