[Math] Conformal mapping of a stripe to upper half of a plane

complex-analysisconformal-geometry

I need to find conformal mapping $W$ which maps half-infinite stripe $Z$, bounded by $2\mathbb{i}$ and $5\mathbb{i}$, to upper half plane.

In other words this is what I have:

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And this is what I want

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I know that $W_1=e^z$ will give me a ring region with radius $e^2$ and $e^5$.

Then my idea was to combine the ring region and the inverse region together to get the full plane: $W_2=(W_1+\frac{1}{W_1})$.

Then I wanted to take only $W_3=Im(W_2)>0$ to get the upper half plane. But I failed to get the analytical expression for $W_3$.

Could you please help me with that? Or maybe I chose the wrong path from the begining?

Best Answer

Hint:

  1. $e^z$ maps the strip $0<Im(z)<\pi$, to the upper half-plane.

  2. Now, translate and shrink the given region to $0<Im(z)<\pi$, $0\leq Re(z)$.

  3. A certain polynomial map will give you the final result.

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