Fatou arc
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
.
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) |
Fatou arc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_arc&oldid=12402