Describe the images under the map $z\mapsto e^z$ of the line $y=x$ I am trying to figure out how to describe the images under the map $z \mapsto e^z$ of the line $y=x$ and of the strip $x-π/2 < y < x+π/2$
I know that $e^z = e^{x+iy}$ and therefore equal to $e^x\cdot (\cos x + i\sin x)$. I was thinking that the image would be a rotation or spiral of some sort?

Your intuition is correct. The equation $e^z=e^{x+iy}=e^xe^{iy}$ allows us to consider $e^z$ in polar form, where the magnitude of $e^z$ is $e^x$ and the the angle is $y$ (since $e^{iy}=\cos(y)+i\sin(y)$ traces out the unit circle in the complex plane as $y$ increases). Thus, the image of the line $y=x$ will be the set of complex numbers whose magnitude is e to the power of their angle. This looks like $r=e^\theta$, which is indeed a spiral.
Similarly, if you consider the strip $x-\pi/2<y<x+\pi/2$, it will look like $e^{\theta-\pi/2}<r<e^{\theta+\pi/2}$, which is a spiral that has increasing thickness, surrounding the original $r=e^\theta$ spiral (as we would expect, since the line $y=x$ is in the center of the strip.
