Find the image of a set under the complex exponential function

Question

How can I find the image of the set {z in C: Re(z)<0, |Im(z)|<π} under the exponential function? It is known that the exponential function maps {x+iy:|y|<π} onto {z:-π< arg(z)< π}, but how can I apply this result in this problem?

Answer 1

For $z\in\Omega$, we may write it as $z=x+iy$, where $x<0,-\pi<y<\pi$.

As a simple try you can consider the image of $z=iy$, though it is not our goal. Notice that $$|e^z|=|e^{iy}|=1$$ for $-\pi<y<\pi.$ This is actually the unit circle without the point $-1$ (since $-1$ is for $y=\pi$ or $-\pi$). Hence, in general, $$|e^z|=|e^{x+iy}|=|e^x|<1$$ since $x<0$. The image will become the open unit disc without the line segment $(-1,0].$ In other words, it is $U\setminus(-1,0]$, where $U$ is the open unit disc.