Conformal map from a region onto the unit disk How to find a conformal one-to-one map from the region given by $|z|<2$ and $|z-1|>1$ onto the unit disk.
Any help will be truly appreciated.
 A: Step 1:Try to map it into a strip. A fractional linear mapping can map a circle to a line. We notice that two boundary curves meet at $(2,0)$, so we set the denominator of the map to $z-2$.So we can map two curves into parallel lines. We try:
$f_1(z)=\frac{z}{z-2}$. We find this map will map the picture you give to a strip between 2 lines: $Rez=0,Rez=1/2$...(1)
Step 2: We must know a conclusion:We can map the upper half plane to unit disc by：
$FROMUPPERTODISC(z)=e^{i\theta}\frac{z-z_0}{z-\bar z_0}$. ($z_0$ is an arbitrary point in upper half plane so we can actually map any point in upper plane to circle center $(0,0)$). We wish to find a way to map the strip (1) to upper plane. First,
$f_2(z)=e^{\frac{i\pi }{2}}$. (1) is mapped into the strip between:
$Imz=0,\,Imz=1/2$...(2)
Step 3:We want to map strip(2) into  upper half plane, let $f_3(z)=2\pi z$. $f_3(z)$ maps strip(2) to strip(3):
$Imz=0*2\pi=0$, $Imz=\frac{1}{2}*2\pi=\pi$...(3)
Step4: Let $f_4(z)=e^z$, we finally map (3) to the upper half plane.
Let $g(z)=\text{FROMUPPERTODISC}(z)\cdot f_4(z)\cdot f_3(z)\cdot f_2(z)\cdot f_1(z)$
$g(z)$ is the desired function.
