Phragmén–Lindelöf principle

In complex analysis, the Phragmén–Lindelöf principle (or method) is a 1908 extension by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf of the maximum modulus principle to unbounded domains.

In the theory of complex functions, it is known that the modulus (absolute value) of a holomorphic (complex differentiable) function in the interior of a bounded region is bounded by its modulus on the boundary of the region. More precisely, if a non-constant function ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ is holomorphic in a bounded region${\displaystyle \Omega }$ and continuous on its closure ${\displaystyle {\overline {\Omega }}=\Omega \cup \partial \Omega }$, then ${\displaystyle |f(z_{0})|<\sup _{z\in \partial \Omega }|f(z)|}$ for all ${\displaystyle z_{0}\in \Omega }$. This is known as the maximum modulus principle. (In fact, since ${\displaystyle {\overline {\Omega }}}$ is compact and ${\displaystyle |f|}$ is continuous, there actually exists some ${\displaystyle w_{0}\in \partial \Omega }$ such that ${\displaystyle |f(w_{0})|=\sup _{z\in \partial \Omega }|f(z)|}$.) The maximum modulus principle is generally used to conclude that a holomorphic function is bounded in a region after showing that it is bounded on the boundary of that region.

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