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Phragmén–Lindelöf principle
In mathematics, the Phragmén–Lindelöf principle is a 1908 extension by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf of the maximum modulus principle of complex analysis, to unbounded domains.
==Background==
In complex function theory it is known that if a function ''f'' is holomorphic in a bounded domain ''D'', and is continuous 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 in the strip 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, 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.
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