Under the transformation $w=z^2,$ find the images of $\arg z=\theta$

Question

please help with this exercise.

Under the transformation $w=z^2,$ find the images of

1. straight line $y=x$

2. "rayo in spanish" $\arg z=\theta$

I try

1- $$K=\{z=x+iy:y=x\}$$

$$w=(x+iy)^2=x^2-y^2+2ixy=2y^2$$

then $u=0$ and $v=2y^2$

$$f(K)=\{w=u+iv: v=2y^2, y\in \mathbb{R}\}$$

2. I don't know how to do it!!!

Answer 1

You have already in the first case that $$y=x\implies u=0,v=2x^2$$ so the locus is the positive imaginary axis.

In the second case, $$\arg w=\arg z^2=2\arg z=2\theta$$ so the locus is the part-line $$\arg w=2\theta$$