View source for Angular boundary value

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''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====
<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====
An angular boundary value is also called a non-tangential boundary value. Cf. [[Boundary properties of analytic functions|Boundary properties of analytic functions]].