# Lindelöf theorem

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on asymptotic values

1) Let be a bounded regular analytic function in the unit disc and let be the asymptotic value of along a Jordan arc situated in and ending at a point , that is, as along . Then is the angular boundary value (non-tangential boundary value) of at , that is, tends uniformly to as inside an angle with vertex formed by two chords of the disc .

The Lindelöf theorem is also true in domains of other types, and the conditions on have been significantly weakened. For example, it is sufficient to require that is a meromorphic function in that does not assume three different values. Lindelöf's theorem can also be generalized to functions of several complex variables . For example, if is a bounded holomorphic function in the ball that has asymptotic value along a non-tangential path at a point , then is the non-tangential boundary value of at (see ).

2) Let be a bounded regular analytic function in the disc that has asymptotic values and along two distinct paths and that end at the point . Then and uniformly inside the angle between the paths and . This theorem is also true for domains 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=18228
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article