Angular boundary value
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.
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