So I'm given a little pizza-slice region with bounds $|z|\leq 2$ and $0\leq y \leq x$
I've already converted this to $r\leq2$ and $0\leq\theta\leq\frac{\pi}{4}$
Now, I'm asked to sketch the image of R under the mapping $w=iz^2$ and explain.
I've never really understood mapping. Would someone be able to give me a walkthrough on this?
I tried looking at $w=iz^2$ so $|w| = i|z|^2$ and from there we get $|z|=\pm\sqrt{\frac{|w|}{i}}$
Then I see that $\pm\sqrt{\frac{|w|}{i}}\leq 2$ so $\sqrt{\frac{|w|}{i}}\leq 2$ and $-\sqrt{\frac{|w|}{i}}\leq2$ Which gets me $|w|\leq4i$ and $|w|\geq4i$ and if $w=u+iv$ then $4i\leq\sqrt{u^2+v^2}\leq4i$ so $\sqrt{u^2+v^2}=4i$ so $u^2+v^2=-16$ but honestly I feel like this is wrong and I have no idea where to go from here even if it's right.
Since $$ R=\{z=re^{i\theta}\;:\;0\le r\le 2, 0\le\theta\le\pi/4\} $$ and writing $i=e^{i\pi/2}$ then calling $\phi(z):=iz^2$, we want to describe $\phi(R)$.
Now if $z\in R$, $$ \phi(z)=\phi(re^{i\theta})=ir^2e^{i2\theta}=r^2e^{i(2\theta+\pi/2)} $$ so, $0\le r \le 2$ clearly implies $0\le r^2\le 4$ and $0\le\theta\le\pi/4$ implies $\pi/2\le2\theta+\pi/2\le\pi$. Now (since these last transformations are bijections in the given domains and codomains) you get that $$ \phi(R)=\{w=se^{i\varphi}\;:\; 0\le s\le4,\;\pi/2\le\varphi\le\pi\} $$ which is a bigger slice (twice the angle and $90°$ rotated) of a bigger pizza (twice the radius).