Find conformal mapping between two sets I am trying to find the conformal mapping between the two sets
$$
D = \lbrace z \in \mathbb{C}: 0 < \Im(z) < \tfrac{\pi}{2} \rbrace,
$$
$$
F = \lbrace z \in \mathbb{C}: \Re(z)>0,  |z| < 1 \rbrace .
$$
My progress so far is to take $D$ to the "first quadrant" of $\mathbb{C}$ by using the mapping $f_1(z) = e^z$. Let's say that the result of $f_1(D) = D_1$.  Now I try to use the mapping $f_2(z) = \frac{z-i}{z+i}$ but I can't visualize the resulting set after doing $f_2(D_1)$.
 A: So we know $f_1(D)=D_1$ is the first quadrant of $\mathbb{C}$. It is the right half of $\mathbb{H}$, itself the upper half-plane of $\mathbb{C}$, and your so-called Cayley transform $g(z)=\frac{z-i}{z+i}$ is well-known to map $\mathbb{H}$ to the unit disk $\mathbb{D}$. Plug in real positive, real negative, and "positive" imaginary $z$ into this, and recognize the formulas for stereographic projection, see $g$ maps the positive real axis to the lower half of the unit circle, the negative real axis to the upper half of the unit circle, and the "positive" imaginary axis to the horizontal diameter of the unit circle. You can then just pick two "test points" in the left and right halves of $\mathbb{H}$ and see the first quadrant goes to the lower half and the second quadrant to the upper half of the unit disk. You can of course multiply by $i$ to turn the lower half into the right half.
Alternatively, instead of using $g$, you can build your own Mobius transformation, using the facts that they preserve the set of generalized circles (i.e. circles and lines) and there is a unique transformation that sends three given points to any other three given points - in particular, the Mobius transformation that sends the three numbers $a,b,c$ to $0,\infty,1$ respectively is the cross-ratio $\frac{z-a}{z-b}\frac{c-b}{c-a}$.
