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.