Represent graphically the set of complex numbers that verifies $|z| = \pi + arg(z)$

Question

This is a problem on my textbook on complex analysis.

Represent graphically the set of complex numbers that verifies the equality: $|z| = \pi + arg(z)$ where arg is the principal value of the argument function.

I've plotted the set with wolframalpha here but I don't know how to do it by hand.

Answer 1

Consider all complex numbers with the same argument $\theta$: $Ae^{i\theta}$. The one that is in your set is when $A = \pi + \theta$.

So the set of numbers has the form $$z = (\pi+\theta)e^{i\theta}$$

And is part of a spiral, subject to the range of $\theta\in(-\pi,\pi]$.

Answer 2

Work in polar coordinates. Write $z=re^{i\theta}$ with $r\ge 0$ and $-\pi<\theta\le\pi$; then the equation in polar coordinates is $r=\theta+\pi$, whose graph is an Archimedean spiral. It’s not hard to plot points by hand for a few nice values of $\theta$ to get a decent sketch.