Fatou arc

From Encyclopedia of Mathematics
Revision as of 16:59, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

for a meromorphic function in a domain of the complex -plane

An accessible boundary arc (cf. Attainable boundary arc) of with the property that it forms part of the boundary of some Jordan domain in which , , is bounded. Sometimes this definition is broadened, replacing the condition that is bounded in by the more general condition that the image of under the mapping is not dense in the -plane. The strengthened version of Fatou's theorem in the theory of boundary properties of analytic functions asserts that if is a Fatou arc (even in the extended sense) for a function that is meromorphic in the disc , then at almost-every point , has a finite limit as tends to from inside within any angle with vertex formed by a pair of chords of .


[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:
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article