- Phragmén-Lindelöf principle
In
mathematics , the Phragmén-Lindelöf principle is a 1908 extension byLars Edvard Phragmén (1863-1937) andErnst Leonard Lindelöf of themaximum modulus principle ofcomplex analysis , to unbounded domains.Background
In complex function theory it is known that if a function "f" is
holomorphic in abounded domain "D", and iscontinuous on the boundary of "D", then the maximum of |"f"| must be attained on the boundary of "D". If, however, the region "D" is not bounded, then this is no longer true, as may be seen by examining the function g(z) = exp(exp(z)) in the strip pi/2 < mbox{Im} { z } < pi/2. The difficulty here is that the function "g" tends to infinity 'very' rapidly as "z" tends to infinity along the positive real axis.The Phragmén-Lindelöf principle shows that in certain circumstances, and by limiting the rapidity with which "f" is allowed to tend to infinity, then it is possible to prove that "f" is actually bounded in the
unbounded domain.In the literature of complex analysis, there are many examples of the Phragmén-Lindelöf principle applied to unbounded regions of differing types, and also a version of this principle may be applied in a similar fashion to subharmonic and superharmonic functions.
Phragmén-Lindelöf principle for a sector in the complex plane
Let F(z) be a function that is
holomorphic in the sector:S = {z : -frac{pi}{4} < arg ,, z < frac{pi}{4}}
and continuous on its boundary. If
:F(z)| leq 1 on the boundary and
:F(z)| leq Ce^{c|z in the sector for some constants c and C, then for all points z in S we have
:F(z)| leq 1.
Phragmén-Lindelöf principle for strips
In practice the point 0 is often transformed into the point ∞ of the
Riemann sphere . This gives a version of the principle that applies to strips, for example bounded by two lines of constantreal part in the complex plane. This special case is sometimes known asLindelöf's theorem .Other special cases
*
Carlson's theorem is an application of the principle to functions bounded on the imaginary axis.References
* cite journal
author = Pharagmén, Lars Edvard and Lindelöf, Ernst
title = Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogénes dans le voisinage d'un point singulier
journal = Acta Math.
volume = 31
year = 1908
issue = 1
pages = 381–406
issn = 0001-5962
doi = 10.1007/BF02415450
* cite journal
last = Riesz
first = Marcel
authorlink= Marcel Riesz
title = Sur le principe de Phragmén-Lindelöf
journal = Proceedings of the Cambridge Philosophical Society
volume = 20
year = 1920 (Corr. cite journal | title = Sur le principe de Phragmén-Lindelöf | volume = 21 | year = 1921 )
* cite book
last = Titchmarsh
first = Edward Charles
authorlink= Edward Charles Titchmarsh
title = The Theory of Functions
edition = Second edition
publisher = Oxford University Press
YEAR = 1976
isbn = 0198533497 (See chapter 5)
*
* cite book
author = Stein, Elias M. and Shakarchi, Rami
title = Complex analysis
series = Princeton Lectures in Analysis, II
publisher = Princeton University Press
location = Princeton, NJ
year = 2003
isbn = 0-691-11385-8
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