8. Special Techniques
Society for Industrial and Applied Mathematics eBooks · 2005
8-1 The Wiener-Hopf MethodThere is an extensive array of important problems whose solution by Fourier-transform methods requires the use of a ingenious technique whose modern usage is due to Wiener and Hopf but which was actually invented by Carleman (cf. Sec. 8-6). The technique is most conveniently displayed in connection with an illustrative example chosen to minimize algebraic complication; accordingly, we study the boundary-value problemΔφ−φx=0inD(8-1)withφ(x,y)→0asx2+y2→∞andφ=e−axony=0,x≥0(8-2)Here a>0 , and D is the domain exterior to the half-line y=0 , x≥0 . We require that ϕ be bounded everywhere and that φy be integrable everywhere.