Finding a conformal map

Question

I'm doing some review for my complex analysis final, and have come across the following. Find a conformal map mapping the half strip : $P=\{x+iy:x<0, 0,<y<\pi\}$ to the upper half plane. I know that conformal means analytic, one to one and angle preserving but don't really know how to define one. I think however it'll involve the exponential function.

Answer 1

$f_1=e^z$ map $P$ to upper half unit disk.

$f_2=\frac{i(1+z)}{1-z}$, inverse Cayley transform map upper half unit disk to first quadrant.

$f_3=z^2$ map first quadrant to upper half plane.

$f_1\circ f_2\circ f_3$ is the map you need.