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The value of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770401.png" />, defined on the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770402.png" />, at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770403.png" />, equal to the 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/r/r077/r077040/r0770404.png" /></td> </tr></table>
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of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770405.png" /> on the set of points of the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770406.png" /> leading to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770407.png" />. The term "radial boundary value" is sometimes used in a generalized sense for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770408.png" /> given on arbitrary (including multi-dimensional) domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704010.png" /> is taken to be the set of points of a normal (or its analogue) to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704011.png" /> leading to the boundary point. For example, in the case of a bi-disc
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The value of a function $  f ( z) $,
 +
defined on the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $,  
 +
at a boundary point  $  \zeta = e ^ {i \theta } $,
 +
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/r/r077/r077040/r07704012.png" /></td> </tr></table>
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$$
 +
\lim\limits _ {r \uparrow 1 }  f ( r e ^ {i \theta } ) \
 +
= f ^ { * } ( e ^ {i \theta } )
 +
$$
  
as the radial boundary value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704013.png" /> one takes the limit
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of the function  $  f ( z) $
 +
on the set of points of the radius  $  H = \{ {z = r e ^ {i \theta } } : {0 < r < 1 } \} $
 +
leading to the point  $  \zeta $.
 +
The term  "radial boundary value" is sometimes used in a generalized sense for functions  $  f ( z) $
 +
given on arbitrary (including multi-dimensional) domains  $  D $,
 +
where  $  H $
 +
is taken to be the set of points of a normal (or its analogue) to the boundary of  $  D $
 +
leading to the boundary point. For example, in the case of a bi-disc
  
<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/r/r077/r077040/r07704014.png" /></td> </tr></table>
+
$$
 +
= \{ {( z _ {1} , z _ {2} ) \in \mathbf C  ^ {2} } : {
 +
| z _ {1} | < 1 , | z _ {2} | < 1 } \}
 +
,
 +
$$
 +
 
 +
as the radial boundary value at  $  \zeta = ( e ^ {i \theta _ {1} } , e ^ {i \theta _ {2} } ) $
 +
one takes the limit
 +
 
 +
$$
 +
\lim\limits _ {r \uparrow 1 } \
 +
f ( r e ^ {i \theta _ {1} } , r e ^ {i \theta _ {2} } )  = \
 +
f ^ { * } ( \zeta ) .
 +
$$
  
 
====References====
 
====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)</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''' , 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====
 
The functions under consideration are usually analytic or harmonic functions. See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]] and its references; cf. also [[Angular boundary value|Angular boundary value]]; and [[Fatou theorem|Fatou theorem]].
 
The functions under consideration are usually analytic or harmonic functions. See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]] and its references; cf. also [[Angular boundary value|Angular boundary value]]; and [[Fatou theorem|Fatou theorem]].

Latest revision as of 08:09, 6 June 2020


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

$$ \lim\limits _ {r \uparrow 1 } f ( r e ^ {i \theta } ) \ = f ^ { * } ( e ^ {i \theta } ) $$

of the function $ f ( z) $ on the set of points of the radius $ H = \{ {z = r e ^ {i \theta } } : {0 < r < 1 } \} $ leading to the point $ \zeta $. The term "radial boundary value" is sometimes used in a generalized sense for functions $ f ( z) $ given on arbitrary (including multi-dimensional) domains $ D $, where $ H $ is taken to be the set of points of a normal (or its analogue) to the boundary of $ D $ leading to the boundary point. For example, in the case of a bi-disc

$$ D = \{ {( z _ {1} , z _ {2} ) \in \mathbf C ^ {2} } : { | z _ {1} | < 1 , | z _ {2} | < 1 } \} , $$

as the radial boundary value at $ \zeta = ( e ^ {i \theta _ {1} } , e ^ {i \theta _ {2} } ) $ one takes the limit

$$ \lim\limits _ {r \uparrow 1 } \ f ( r e ^ {i \theta _ {1} } , r e ^ {i \theta _ {2} } ) = \ f ^ { * } ( \zeta ) . $$

References

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

Comments

The functions under consideration are usually analytic or harmonic functions. See also Boundary properties of analytic functions and its references; cf. also Angular boundary value; and Fatou theorem.

How to Cite This Entry:
Radial boundary value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radial_boundary_value&oldid=48410
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article