# 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

 [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