Namespaces
Variants
Actions

Difference between revisions of "Lindelöf theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Lindelof theorem to Lindelöf theorem over redirect: accented title)
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
l0590001.png
 +
$#A+1 = 45 n = 1
 +
$#C+1 = 45 : ~/encyclopedia/old_files/data/L059/L.0509000 Lindel\AGof theorem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''on asymptotic values''
 
''on asymptotic values''
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590001.png" /> be a bounded regular analytic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590002.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590003.png" /> be the [[Asymptotic value|asymptotic value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590004.png" /> along a Jordan arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590005.png" /> situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590006.png" /> and ending at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590007.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590008.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590009.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900011.png" /> is the [[Angular boundary value|angular boundary value]] (non-tangential boundary value) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900013.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900014.png" /> tends uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900016.png" /> inside an angle with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900017.png" /> formed by two chords of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900018.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900019.png" /> of other types, and the conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900020.png" /> have been significantly weakened. For example, it is sufficient to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900021.png" /> is a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900022.png" /> that does not assume three different values. Lindelöf's theorem can also be generalized to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900023.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900024.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900025.png" /> is a bounded holomorphic function in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900026.png" /> that has asymptotic value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900027.png" /> along a non-tangential path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900028.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900030.png" /> is the non-tangential boundary value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900032.png" /> (see [[#References|[4]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900033.png" /> be a bounded regular analytic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900034.png" /> that has asymptotic values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900036.png" /> along two distinct paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900038.png" /> that end at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900039.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900041.png" /> uniformly inside the angle between the paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900043.png" />. This theorem is also true for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900044.png" /> of other types. For unbounded functions it is false, generally speaking.
+
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]]].
 
These theorems were discovered by E. Lindelöf [[#References|[1]]].
Line 11: Line 66:
 
====References====
 
====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>
 
<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====
 
====Comments====
For the generalization of Lindelöf's theorem to functions of several variables, the condition that the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900045.png" /> is non-tangential may be weakened, see [[#References|[a1]]], Chapt. 8.
+
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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900046.png" />" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900046.png" />" , Springer  (1980)</TD></TR></table>

Revision as of 22:16, 5 June 2020


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 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 (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 [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 [1].

References

[1] 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
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[4] 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

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 [a1], Chapt. 8.

References

[a1] W. Rudin, "Function theory in the unit ball in " , Springer (1980)
How to Cite This Entry:
Lindelöf theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_theorem&oldid=23396
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article