Image sets in Complex Analysis I was given the problem "what is the image set of the first quadrant in the $z$ plane under the mapping $w=z^4$, but I have no idea how to even think about this.
The most I did was write $w=|z|^4 (\cos(4\theta) +i\sin(4\theta))$
But again, Im not sure if that's helpful or not. How do we picture things like this?

Yes, writing your complex number $z$ in polar form is a good start.  But now think about what raising a complex number to the fourth power means: the magnitude gets raised to the fourth power and the angle gets multiplied by four.  So as the angle of $z$ ranges in $[0,\frac{\pi}{2}]$, the angle of $z^4$ ranges in $[0,2\pi]$.  Now think about what happens when we take an arbitrary non-negative real number and raise it to the fourth power...
