View source for Radial boundary value

The value of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770401.png" />, defined on the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770402.png" />, at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770403.png" />, equal to the limit

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of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770405.png" /> on the set of points of the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770406.png" /> leading to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770407.png" />. The term  "radial boundary value"  is sometimes used in a generalized sense for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770408.png" /> given on arbitrary (including multi-dimensional) domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r0770409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704010.png" /> is taken to be the set of points of a normal (or its analogue) to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704011.png" /> leading to the boundary point. For example, in the case of a bi-disc

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as the radial boundary value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077040/r07704013.png" /> one takes the limit

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====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]].