I'm having trouble finding the range of $z\mapsto z^4 | \Re(z) + \Im(z) ≥0\text{ and }\Re(z)< 0 $. If I set $z=x+iy$ and let $x+y=0$, I get: $z^4= (x+iy)^2 = (x-ix)^4 = x^4(1-i)^4 = x^4(-2i)^2 = -4x^4$.
Since $x$ is real, this tells me the line $x+y=0$ maps to the negative real axis. What I have trouble doing now is filling the rest of the region. Since $\Re(z) + \Im(z) ≥0$ and $\Re(z)<0$ represents all complex numbers between of argument between $\frac{3\pi}{4}$ and $\frac{\pi}{2}$, then $z^4$ would have arguments between $3\pi$ and $2\pi$ which looks like the top half of the plane. I however can't decide on its shape or what it looks like. Does it really just cover the entire half of the plane? If so, how do I show that? In sum, I am looking for a more systematic way of determining the rest of this mapping so that I can apply it to other problems of this type, thanks!
