# 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 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 ).

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 .

How to Cite This Entry:
Lindelöf theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_theorem&oldid=47644
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article