View source for Fatou arc

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''for a meromorphic function $f(z)$ in a domain $G$ of the complex $z$-plane''

An accessible boundary arc (cf. [[Attainable boundary arc|Attainable boundary arc]]) of $G$ with the property that it forms part of the boundary of some Jordan domain $g\subset G$ in which $f(z)$, $z\in\mathbf C$, is bounded. Sometimes this definition is broadened, replacing the condition that $f(z)$ is bounded in $g$ by the more general condition that the image of $g$ under the mapping $w=f(z)$ is not dense in the $w$-plane. The strengthened version of Fatou's theorem in the theory of boundary properties of analytic functions asserts that if $\gamma$ is a Fatou arc (even in the extended sense) for a function $f(z)$ that is meromorphic in the disc $D=\{|z|<1\}$, then at almost-every point $\zeta\in\gamma$, $f(z)$ has a finite limit as $z$ tends to $\zeta$ from inside $D$ within any angle with vertex $\zeta$ formed by a pair of chords of $D$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</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><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

[[Category:Functions of a complex variable]]