Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 86 / 3 / a / Solution

Let $s_j=\sin {\beta }_j$, $c_j=\cos {\beta }_j$. Inverting the two orthogonal derivative systems gives, on the vertical axis,and, on the horizontal axis,Using the Wirtinger derivative $q_z=(q_x-i{q}_y)/2$, these become

Define the appropriate boundary transforms byThe first pair is holomorphic for $\mathrm{Re}k<0$; the second pair consists of Half-range Fourier transforms, holomorphic for $\mathrm{Im}k<0$. Parametrizing the vertical boundary by $z=iy$ gives $dz=idy$, which must be retained. Substitution yields the quarter-plane oblique boundary spectral functions:Their signs follow from the stated clockwise boundary orientations: upward on the left and inward on the bottom.