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''for a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382401.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382402.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382403.png" />-plane''
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382404.png" /> with the property that it forms part of the boundary of some Jordan domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382405.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382407.png" />, is bounded. Sometimes this definition is broadened, replacing the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382408.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f0382409.png" /> by the more general condition that the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824010.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824011.png" /> is not dense in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824012.png" />-plane. The strengthened version of Fatou's theorem in the theory of boundary properties of analytic functions asserts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824013.png" /> is a Fatou arc (even in the extended sense) for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824014.png" /> that is meromorphic in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824015.png" />, then at almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824017.png" /> has a finite limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824018.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824019.png" /> from inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824020.png" /> within any angle with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824021.png" /> formed by a pair of chords of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038240/f03824022.png" />.
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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====
 
====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>
 
<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>

Revision as of 07:12, 15 July 2014

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=12402
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article