Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 61 / 2 / a / Solution

Write $\chi =e^{-ikz-\lambda \bar{z}/(ik)}$. In terms of Wirtinger derivatives, the elliptic equation is $q_{z\bar{z}}=\lambda q$. If $W=Adz+Bd\bar{z}$, its coefficients are $A=\chi q_z$ and $B=-\lambda \chi q/(ik)$. Direct differentiation givesTherefore $dW=(B_z-A_{\bar{z}})dz\wedge d\bar{z}=0$, proving that $W$ is a closed differential one-form. The parameter restriction $k\ne 0$ makes its coefficients well defined.