Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 3 / b

Use the principal complex logarithm:The excluded nonpositive real ray allows a continuous choice of argument, and $L(1)=0$. For $z=x+iy$, its real part is $u=\frac{1}{2}\log (x^2+y^2)$. Differentiating the chosen local argument givesThese continuous partial derivatives satisfy the Cauchy-Riemann equations, proving holomorphy on the slit domain. Its derivative is

A holomorphic primitive on the slit domain is $P(z)=zL(z)-z$, since the product rule and $L^{\prime }=1/z$ give $P^{\prime }=L$. The line segment stays in that domain. The fundamental theorem for line integrals therefore givesHenceThe value $L(i)=\pi i/2$ is fixed by the chosen branch.

In the upper half-plane, the argument lies strictly between zero and $\pi$, while $\log |z|$ can be any real number. Conversely, if $w=s+it$ with $0<t<\pi$, then $z=e^w$ is in the upper half-plane and $L(z)=w$.Thus the principal complex logarithm is a conformal bijection from the upper half-plane onto that horizontal strip, with inverse the exponential function restricted to the strip.