Radial boundary value

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.