Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 7 / v

Assume the image of the domain of $f$ lies in the domain where $g$ is holomorphic. The complex chain rule givesso the composition is holomorphic, including at points where $f^{\prime }=0$. Its real part $S(x,y)=s(u(x,y),v(x,y))$ is therefore a harmonic function.

A direct verification uses the composition of harmonic conjugate pairs. The multivariable chain rule givesThe Cauchy-Riemann equations imply $|\nabla u{|}^2=|\nabla v{|}^2$; parts (i) and (ii) remove the last three terms. Since $s$ is harmonic in its own variables,