Wiener–Hopf method: Difference between revisions

Revision as of 12:54, 19 May 2020

Revision as of 13:14, 19 May 2020

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where the contours <math>C_1</math> and <math>C_2</math> are parallel to the real line, but pass above and below the point <math>z=\alpha</math>, respectively.

Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. <math>K(\alpha) = K_+(\alpha)K_-(\alpha)</math>, by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.{{citation needed|date=May 2020}}

==Example==