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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>

====Comments====
The functions under consideration are usually analytic or harmonic functions. See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]] and its references; cf. also [[Angular boundary value|Angular boundary value]]; and [[Fatou theorem|Fatou theorem]].