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As the title states, I need to find a conformal mapping from the vertical strip $\{-1 < Re(z) < 1\}$ onto open unit disk $D$. I'm not sure how to do this.. so do I first have to find a few points? For example, do I want the line $Re(z) = 1$ to map to (0,0) and then $Re(z) = -1$ to map to the $|z| = 1$?

Thanks!

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A conformal map is a one-to-one and onto analytic map, so you cannot map $Re(z) = 1$ to a point as you said.

Here is the sketch. First, let $f_1$ be defined by $f_1(z)=\frac{i\pi}{2}(z+1)$, which is a conformal mapping from the strip $|Rez|<1$ onto the strip $0<Im(z)<i\pi$ by translation, rotation and scalar multiplication. Let $f_2$ be the exponential map, which is a conformal mapping from $0<Im(z)<i\pi$ onto the upper half plane. Let $f_3$ be defined by $f_3(z)=\frac{z-i}{z+i}$, which is a conformal mapping from the upper half plane onto the unit disk.

Therefore, $f_3\circ f_2\circ f_1$ is the desired conformal mapping.

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