Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 86 / 3 / a / Solution

Write $k=r{e}^{i\theta }$, with $r>0$, and rotate the plane into the line coordinatesThus $x_1=\tau \cos \theta -\rho \sin \theta$ and $x_2=\tau \sin \theta +\rho \cos \theta$, agreeing with the directed-line geometry in the PDF. Direct substitution, rather than a formal complex rotation, gives the directional differential operatorwhere $A=(r+r^{-1})/2$ and $B=(r^{-1}-r)/2$. If $r\ne 1$, choose the invertible real-linear complex coordinateThe Wirtinger derivative in this coordinate is ${\partial }_{\bar{z}}=\frac{1}{2}[{\partial }_{\tau }+i(B/A){\partial }_{\rho }]$, so $D_k=\nu (r){\partial }_{\bar{z}}$. Substitution gives the required transformed equation with the same multiplication and source terms. This $z$ is a new spectral-dependent spatial coordinate, not the physical $x_1+i{x}_2$ used in Question 2.

The complex change of variables is valid for $0<|k|\ne 1$. At $|k|=1$, $B=0$ and the operator becomes the real line derivative ${\partial }_{\tau }$; there can be no invertible elliptic complex-coordinate change there. The two limits to this characteristic circle are precisely the boundary values needed in part (b). The expression at $k=0$ is undefined in the printed equation and is treated through its spectral limit, not by substitution.