Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 7 / b / Solution

For the real part specified in the PDF,Integrating $v_y=u_x$ givesSubstitution into $v_x=-u_y$ shows $h^{\prime }(x)=0$. ThusIndeed $\cos (x+iy)=\cos x\cosh y-i\sin x\sinh y$, so the real part of $z\cos z$ is exactly the prescribed expression. The arbitrary imaginary constant is the only freedom on a connected domain.