I am interested in finding explicit formulae for (better yet characterizing) conformal functions from various domains onto the open unit disc $\mathbb{D}\subset\mathbb{C}$, and in understanding the key ideas necessary to establish such functions.
Specifically, what can $f$ look like when $f:G\to\mathbb{D}$ is conformal and
(1) $G=\{x+iy~|~x,y>0\}$ is the open first quadrant.
(2) $G=\{x+iy~|~x>0,~0<y<1\}$ is an open horizontal strip in the first quadrant.
(3) $G=\{z\in\mathbb{C}~|~\frac{1}{2}<|z|<1\}$ is an annulus.
(4) $G=\mathbb{D}\cap\{|z-\frac{1}{2}|>\frac{1}{2}\}$ is something else (torus?).
Best Answer
Since you didn't show too many own thoughts, here are some hints only. By conformal I understand biholomorphic.
First take $f(z) = z^2$ to map the quadrant biholomorphically onto the upper half-plane, then compose with the Cayley transform $\kappa(z) = \frac{z-i}{z+i}$ to get $\kappa(f(z)) = \frac{z^2-i}{z^2+i}$.
Look at $\cos{(z)}$ and modify appropriately.
Impossible, since $G$ is not simply connected.
Map the region $G$ to the strip between two parallel lines using a Möbius transformation sending $1$ to infinity (e.g. using the inverse Cayley transformation). Then use the exponential function.
This should be enough to figure the solutions out.
For the precise relationship between "conformal" and "analytic", as well as for explanations on how to find such maps, I refer you to Ahlfors or (probably—I never really read it) Needham or any decent text on complex analysis treating conformal mapping.
The characterization of biholomorphisms between simply connected regions is essentially the content of the Riemann mapping theorem.
Sometimes biholomorhic mappings between polygonal regions and the unit disk can be computed via the Schwarz-Christoffel formula, but usually it leads to elliptic integrals that can't be solved explicitly in elementary terms.
Added:
Since the solution of 4. is a bit trickier, here's a rather detailed outline:
First note that $G$ is the region enclosed between the circles $\{|z| = 1\}$ and $\{|z - \frac{1}{2}| = \frac{1}{2}\}$. Applying the Möbius transformation (= the inverse Cayley transform) $\kappa^{-1}(z) = i\frac{1+z}{1-z}$ sends $G$ to the horizontal strip $\{0 \lt \operatorname{Im}{z} \lt 1\}$. To see this, look at this picture from Wikipedia illustrating the Cayley transform:
Finally, the exponential function $g(z) = \exp{(\pi z)}$ sends this strip to the upper half plane. Composing this with the Cayley transform we get the biholomorphic map $h = \kappa \circ g \circ \kappa^{-1}: G \to \mathbb{D}$.