Phragmén–Lindelöf principle (Redirected from Phragmén–Lindelöf theorem) Jump to navigation Jump to search In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function {displaystyle f} (i.e, {displaystyle |f(z)|0} and {displaystyle |fh_{epsilon }|leq M} on the boundary {displaystyle partial S_{mathrm {bdd} }} of an appropriate bounded subregion {displaystyle S_{mathrm {bdd} }subset S} ; and (ii): the asymptotic behavior of {displaystyle fh_{epsilon }} allows us to establish that {displaystyle |fh_{epsilon }|leq M} for {displaystyle zin Ssetminus {overline {S_{mathrm {bdd} }}}} (i.e., the unbounded part of {displaystyle S} outside the closure of the bounded subregion). This allows us to apply the maximum modulus principle to first conclude that {displaystyle |fh_{epsilon }|leq M} on {displaystyle {overline {S_{mathrm {bdd} }}}} and then extend the conclusion to all {displaystyle zin S} . Finally, we let {displaystyle epsilon to 0} so that {displaystyle f(z)h_{epsilon }(z)to f(z)} for every {displaystyle zin S} in order to conclude that {displaystyle |f|leq M} on {displaystyle S} .
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.
Example of application To continue the example above, we can impose a growth condition on a holomorphic function {displaystyle f} that prevents it from "blowing up" and allows the Phragmén–Lindelöf principle to be applied. To this end, we now include the condition that {displaystyle |f(z)|0} the auxiliary function {displaystyle h_{epsilon }} by {textstyle h_{epsilon }(z)=e^{-epsilon (e^{bz}+e^{-bz})}} . Moreover, for a given {displaystyle a>0} , we define {displaystyle S_{a}} to be the open rectangle in the complex plane enclosed within the vertices {displaystyle {apm ipi /2,-apm ipi /2}} . Now, fix {displaystyle epsilon >0} and consider the function {displaystyle fh_{epsilon }} . It can be shown that {displaystyle |f(z)h_{epsilon }(z)|to 0} as {displaystyle |Re (z)|to infty } . This allows us to find an {displaystyle x_{0}} such that {displaystyle |f(z)h_{epsilon }(z)|leq 1} whenever {displaystyle zin {overline {S}}} and {displaystyle |Re (z)|geq x_{0}} . Because {displaystyle S_{x_{0}}} is a bounded region, and {displaystyle |f(z)h_{epsilon }(z)|leq 1} for all {displaystyle zin partial S_{x_{0}}} , the maximum modulus principle implies that {displaystyle |f(z)h_{epsilon }(z)|leq 1} for all {displaystyle zin {overline {S_{x_{0}}}}} . Since {displaystyle |f(z)h_{epsilon }(z)|leq 1} whenever {displaystyle zin S} and {displaystyle |Re (z)|>x_{0}} , {displaystyle |f(z)h_{epsilon }(z)|leq 1} in fact holds for all {displaystyle zin S} . Finally, because {displaystyle fh_{epsilon }to f} as {displaystyle epsilon to 0} , we conclude that {displaystyle |f(z)|leq 1} for all {displaystyle zin S} . Q.E.D.
Phragmén–Lindelöf principle for a sector in the complex plane A particularly useful statement proved using the Phragmén–Lindelöf principle bounds holomorphic functions on a sector of the complex plane if it is bounded on its boundary. This statement can be used to give a complex analytic proof of the Hardy's uncertainty principle, which states that a function and its Fourier transform cannot both decay faster than exponentially.[3] Proposition. Let {displaystyle F} be a function that is holomorphic in a sector {displaystyle S=left{z,{big |},alpha 0} , then (1) holds also for all {displaystyle zin S} .
Remarks The condition (2) can be relaxed to {displaystyle liminf _{rto infty }sup _{alpha
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