# Fatou arc

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

An accessible boundary arc (cf. 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

[1] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6 |

[2] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

[3] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |

**How to Cite This Entry:**

Fatou arc.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fatou_arc&oldid=34174