Past exam of the applied complex variables course of the University of Oxford / 2011 / 3 / b / i / Solution

Write $Y_{\pm }={\overline{y}}_{\pm }$, $y_0=y(0)$ and $d_{\pm }=y^{\prime }(0\pm )$. In a common convergence strip, two integrations by parts on each half-line giveThe boundary terms at infinity vanish by exponential decay; the ordinary differential equations also give decay of the required first derivatives. Substituting the two equations for $y^{\prime \prime }$ yieldsContinuity makes the $y_0$ terms cancel on addition, while the unit derivative jump givesMultiply by $-1/((k-a)(k+b))$ and factor the quadratics. Since $\mathrm{Im}(a+b)>0$, their sum is nonzero, and partial fractions giveThis is the Wiener-Hopf equation for the two half-line Fourier transforms. The plus and minus subscripts refer to support in physical space; their holomorphic domains in transform space are determined next.