# Sketching Domains and Images in Complex Analysis

Sketch the domain $S:=\{x+iy:\;x<0,\;\pi/4<y\leq\pi/2\}$ and its image $T$ under the exponential function.

My question is how do I begin to think about this region? I think we can change $z=re^{i\theta}$, where $\pi/4<\theta\leq\pi/2$, I don't even know where this will take me.

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The Maple commands $$with(plots):conformal(exp(z), z = -10^9+(1/4)*Pi*I .. 0+(1/2)*Pi*I, scaling = constrained)$$ produce a very good approximation of $T$. – user64494 Aug 26 '13 at 4:35

Consider $z = x + iy \in S$, and $T(z)$:

$$T(z) = e^{x + iy} = e^x e^{iy}$$

Now since $y \in (\pi/4, \pi/2]$, this means that the argument of $T(z)$ lies between $\pi/4$ and $\pi/2$, including the second number. Similarly, the modulus is

$$|T(z)| = e^x < 1$$

So the image is the part of the unit disk whose angle lies in $(\pi/4, \pi/2]$.

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Okay the math makes sense. Now in layman's terms what is the statement asking? – Mr.Fry Aug 26 '13 at 3:19
It's asking: If you "exponentiate a strip," what region do you get? One decent way to get intuition is to pick some points in the strip, and explicitly compute what $T$ does to them. These maps will generally stretch and bend the region, and watching specific points can help you see exactly how that happens. – user61527 Aug 26 '13 at 3:23