Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 1 / 7B (Complex Methods)

Since $u$ and $v$ are harmonic conjugates, $f=u+iv$ is a holomorphic function on $D$, with $u_x=v_y$ and $u_y=-v_x$. The entire function $w\mapsto e^{i{w}^2}$ can be composed with $f$. Its value isBy the complex chain rule the composition is holomorphic on $D$, so its real and imaginary parts obey the Cauchy-Riemann equations. They are harmonic as well: differentiating those equations gives $U_{xx}+U_{yy}=V_{yx}-V_{xy}=0$ and likewise $V_{xx}+V_{yy}=0$. Hence $U,V$ are a pair of harmonic conjugates on $D$. No simple-connectivity assumption is needed because the conjugate pair is already supplied globally. The PDF has $e^{-2uv}$ in both expressions; the differing TeX exponents are transcription defects.