View source for Lindelöf theorem

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''on asymptotic values''

1) Let  $  w = f ( z) $
be a bounded regular analytic function in the unit disc  $  D = \{ {z } : {| z | < 1 } \} $
and let  $  \alpha $
be the [[Asymptotic value|asymptotic value]] of  $  f ( z) $
along a Jordan arc  $  L $
situated in  $  D $
and ending at a point  $  e ^ {i \theta _ {0} } $, 
that is,  $  f ( z) \rightarrow \alpha $
as  $  z \rightarrow e ^ {i \theta _ {0} } $
along  $  L $. 
Then  $  \alpha $
is the [[Angular boundary value|angular boundary value]] (non-tangential boundary value) of  $  f ( z) $
at  $  e ^ {i \theta _ {0} } $, 
that is,  $  f ( z) $
tends uniformly to  $  \alpha $
as  $  z \rightarrow e ^ {i \theta _ {0} } $
inside an angle with vertex  $  e ^ {i \theta _ {0} } $
formed by two chords of the disc  $  D $.

The Lindelöf theorem is also true in domains  $  D $
of other types, and the conditions on  $  f ( z) $
have been significantly weakened. For example, it is sufficient to require that  $  f ( z) $
is a meromorphic function in  $  D $
that does not assume three different values. Lindelöf's theorem can also be generalized to functions  $  f ( z) $
of several complex variables  $  z = ( z _ {1} \dots z _ {n} ) $. 
For example, if  $  f ( z) $
is a bounded holomorphic function in the ball  $  D = \{ {z } : {| z | < 1 } \} $
that has asymptotic value  $  \alpha $
along a non-tangential path  $  L $
at a point  $  \zeta \in \partial  D $, 
then  $  \alpha $
is the non-tangential boundary value of  $  f ( z) $
at  $  \zeta $(
see [[#References|[4]]]).

2) Let  $  w = f ( z) $
be a bounded regular analytic function in the disc  $  D = \{ {z } : {| z | < 1 } \} $
that has asymptotic values  $  \alpha $
and  $  \beta $
along two distinct paths  $  L _ {1} $
and  $  L _ {2} $
that end at the point  $  e ^ {i \theta _ {0} } $. 
Then  $  \alpha = \beta $
and  $  f ( z) \rightarrow \alpha $
uniformly inside the angle between the paths  $  L _ {1} $
and  $  L _ {2} $. 
This theorem is also true for domains  $  D $
of other types. For unbounded functions it is false, generally speaking.

These theorems were discovered by E. Lindelöf [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,   "Sur un principe générale de l'analyse et ses applications à la théorie de la représentation conforme"  ''Acta Soc. Sci. Fennica'' , '''46''' :  4  (1915)  pp. 1–35</TD></TR><TR><TD valign="top">[2]</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><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  E.M. [E.M. Chirka] Čirka,   G.M. [G.M. Khenkin] Henkin,   "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5'''  (1976)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovrem. Probl.'' , '''4'''  (1975)  pp. 13–142</TD></TR></table>

====Comments====
For the generalization of Lindelöf's theorem to functions of several variables, the condition that the path  $  L $
is non-tangential may be weakened, see [[#References|[a1]]], Chapt. 8.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in $\CC^n$" , Springer  (1980)</TD></TR></table>