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Radial boundary value

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The value of a function , defined on the unit disc , at a boundary point , equal to the limit

of the function on the set of points of the radius leading to the point . The term "radial boundary value" is sometimes used in a generalized sense for functions given on arbitrary (including multi-dimensional) domains , where is taken to be the set of points of a normal (or its analogue) to the boundary of leading to the boundary point. For example, in the case of a bi-disc

as the radial boundary value at one takes the limit

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.

How to Cite This Entry:
Radial boundary value. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Radial_boundary_value&oldid=15727
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098