The reciprocal transformation

Question

Show that $W(z)=\frac{1}{z}$ maps the interior of the unit circle in the first quadrant to the exterior of the unit circle in the fourth quadrant.

I can see why it does this but I can't explain it clearly and mathematically.

Answer 1

Let $z$ be a point in the interior of the unit circle in the first quadrant. We can write \begin{equation} z = e^re^{i\theta}, \end{equation}

where necessarily we have $r<0$ and $0\leqslant \theta < \pi/2$. Substitution into $W(z)$ then yields \begin{equation} W(z) = \frac{1}{e^re^{i\theta}} = e^{-r}e^{-i\theta} = e^se^{i\tau}, \end{equation}

where $s=-r>0$ and $\tau = 2\pi - \theta$, so that $\frac{3\pi}{4} \leqslant \tau < 2\pi$. Hence $W(z)$ is in the exterior of the unit circle in the fourth quadrant.