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  "body": "\n## Summary\n\nThe Wiener–Hopf method solves [[Wiener–Hopf equations|QYY4NMDY5H]] and boundary value problems whose conditions change type across a half-line.\nIt takes [[Fourier transforms|DPVM34ADZV]], so that functions supported on a half-line become functions holomorphic in a half-plane.\nThe key step factors the symbol of the problem into two factors.\nOne is holomorphic and nonzero in the upper half-plane, the other in the lower half-plane.\nAn argument with [[Liouville's theorem|0EHKVRF4J6]] then pins down the solution.\nIt was introduced by Norbert Wiener and Eberhard Hopf for integral equations on a half-line.\nIt is a standard tool in diffraction theory, radiative transfer and fracture mechanics.\n\n## Explanations\n\n### L1 human\n\nA function that vanishes on the negative half-line has a Fourier transform that extends nicely to the upper half of the complex plane.\nA function that vanishes on the positive half-line extends to the lower half.\nThe method rewrites the equation so that one side extends to the upper half-plane and the other side to the lower half-plane.\nSince both sides are equal on the real line, together they form one function on the whole plane, which is then a polynomial by Liouville's theorem.\nReading off this polynomial gives the solution.\n",
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