Use conformal maps or combinations of conformal maps such as linear fractional transformations, powers, roots, sin z, log z, etc., to find a one-to-one analytic function mapping the given region D onto the upper half-plane U.

(1) $\displaystyle D = \{z: |Arg z| < \alpha\}, \alpha \leq \pi$

(2) $\displaystyle D = \{z= x + iy: x,y > 0\}$

(3) $\displaystyle D = \{z= x+ iy: |y-1| < 2\}$

(4) $\displaystyle D = \{z: |z-z_0| < r_0\}$

I'll try to do (1)

$\displaystyle -\alpha < Arg z < \alpha$

$\displaystyle z=re^{i\gamma}$

$\displaystyle z^{\alpha}=r^{\alpha}e^{i\theta\alpha}$

$\displaystyle -\alpha\gamma < \theta\gamma <\alpha\gamma$

$\displaystyle 2\alpha\gamma = \pi$

$\displaystyle \gamma = \frac{\pi}{2\alpha}$

Answer: $\displaystyle iz^{\frac{\pi}{2\alpha}}$

If anyone could tell me if I'm on the right track and provide me with some help on the other problems, I would really appreciate it, thanks!