Wiener–Hopf method: Difference between revisions

Revision as of 10:04, 13 February 2024

Latest revision as of 22:48, 18 July 2025

(One intermediate revision by one other user not shown)

Line 5:

== Wiener–Hopf decomposition ==

The fundamental equation that appears in the Wiener-Hopf method is of the form

The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math>\Phi</math> into two functions <math>\Phi_{\pm}</math> with the desired properties outlined above. In general, this can be done by writing

:<math>A(\alpha)\Xi_+(\alpha) + B(\alpha)\Psi_-(\alpha) + C(\alpha) =0, </math>

where <math>A</math>, <math>B</math>, <math>C</math> are known [[holomorphic function]]s, the functions <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> are unknown and the equation holds in a strip <math>\tau_- < \mathfrak{Im}(\alpha) < \tau_+</math> in the [[Complex_plane|complex <math>\alpha</math> plane]]. Finding <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> is what's called the '''Wiener-Hopf problem'''.{{sfn | Noble | 1958 | loc=§4.2}}

The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math>\Phi</math> into two functions <math>\Phi_{\pm}</math> with the desired properties outlined above. In general, this can be done by writing

: <math>\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1} \Phi(z) \frac{dz}{z-\alpha}</math>

Line 75 ⟶ 79:

* {{Cite web|title=Category:Wiener-Hopf - WikiWaves|url=https://wikiwaves.org/Category:Wiener-Hopf|website=wikiwaves.org|access-date=2020-05-19}}

* {{SpringerEOM |id=W/w097910|title=Wiener-Hopf method}}

* {{Cite book|last=Fornberg, Bengt|url=https://www.worldcat.org/oclc/1124781689|title=Complex variables and analytic functions : an illustrated introduction|others=Piret, Cécile.|date=2020 |isbn=978-1-61197-597-0|___location=Philadelphia|oclc=1124781689}}

* {{cite book | last=Noble | first=Ben | title=Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations | publisher=Taylor & Francis US | publication-place=New York, N.Y | date=1958 | isbn=978-0-8284-0332-0}}