Complex Analysis: Theory and Applications · Problem 5.2.12

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · complex-logarithm-mapping

Problem

Show that the mapping

w = \mathrm{Log}(z)

maps ((a)) the ray

\{z = re^{i\theta} : r > 0,\ \theta = \frac{\pi}{3}\}

one-to-one and onto the horizontal line

\{(u,v) : v = \frac{\pi}{3}\}

. ((b)) the semicircle

\{z = 2e^{i\theta} : -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\}

one-to-one and onto the vertical line segment

\{(\ln 2, v) : -\frac{\pi}{2} \leq v \leq \frac{\pi}{2}\}