Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3

For a complex derivative to exist, the difference quotient must agree along real and imaginary increments. These give $u_x+i{v}_x$ and $v_y-i{u}_y$, respectively. Equating their real and imaginary parts gives the Cauchy-Riemann equationsFor $g(z)=z\mathrm{Re}z$, the components are $u=x^2$ and $v=xy$. The first equation requires $2x=x$, so $x=0$; the second requires $0=-y$, so $y=0$. Thus zero is the only possible differentiability point. Directly,so $\boxed{g^{\prime }(0)=0}$. The function is differentiable at that isolated point but is not holomorphic on any neighborhood of it.

Use the chain rule with $x=r\cos \theta$, $y=r\sin \theta$. Then $U_r=u_x\cos \theta +u_y\sin \theta$ and $V_{\theta }=-r{v}_x\sin \theta +r{v}_y\cos \theta$. Substituting the Cartesian Cauchy-Riemann equations gives $V_{\theta }=r{U}_r$. Similarly $V_r=v_x\cos \theta +v_y\sin \theta =-U_{\theta }/r$. Thus, for $r>0$, the polar equations areThe restriction $r>0$ is needed because the polar chart degenerates at the origin.

On the unit circle use $z=e^{i\theta }$ with $0<\theta <2\pi$. The specified branch has $\mathrm{Log}z=i\theta$ there and $dz=i{e}^{i\theta }d\theta$. ThereforeThis is the circle integral of a logarithm with a radial branch cut, using the one-sided values at the cut crossing. The logarithm is discontinuous at that crossing, so the Cauchy integral theorem for a function holomorphic around a closed contour does not apply.

Choose the argument interval $-3\pi /2<\theta <\pi /2$, which makes the logarithm real on the positive real axis and puts its cut on the positive imaginary axis. Parametrize $z=R{e}^{i\theta }$. The constant term $\ln R$ contributes $\ln R\oint dz=0$, leavingEquivalently the cut-crossing point is $iR$, so the circle integral of a logarithm with a radial branch cut is $2\pi i(iR)=-2\pi R$. The different cut changes where the discontinuity lies on the circle; it is not merely an additive constant change of branch along the entire contour.