Sketch this subset in the complex plane

I'm trying to plot this in the complex plane:

$$C = \{z \in\mathbb C | z \neq 0,\arg(z^2) \in \left[0, \pi/4\right)\}$$

My work so far: Let $$z = re^{(i\theta)}$$ $$z^2 = r^2(\cos(2\theta) + i\sin(2\theta))$$

I know how to plot in the complex plane, but I'm not really sure how to specifically plot this function. Thanks in advance for your help!

• You write $z=0$ and that does not leave you many choices... – dmtri Sep 22 '18 at 17:58

2 Answers

You want $$2\theta$$ between zero and $$\pi/4$$ thus $$\theta$$ is between zero and $$\pi/8$$

That is the part of the complex plane between the two rays $$\theta =0$$ and $$\theta =\pi/8$$

• ahh makes sense, thank you! :) – Chinmayee Gidwani Sep 22 '18 at 18:33

Having $$z^2=e^{i\theta}$$, must be $$z=e^{i\theta/2}$$ So,

$$C=\{z\in\mathbb C|\arg(z)\in\left[0,\pi/8\right)\}$$

It's the angle of value $$\pi/8$$, one side along $$\mathbb R^+$$, the other along the ray of complex numbers with $$\pi/8$$ and (consequently) vertex in $$0$$.