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.
Conformal map from a region onto the unit disk
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complex-analysis
1 Answers
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Step 1:Try to map it into a strip.Since a fractional linear mapping can map a circle to a line,we can try it.We notice that two curves meet at $(2,0)$,so we Let the denominator of the map to be $z-2$.So we can map two curves into parallel lines(Since 2 curves inscribe).We don't know the numerator,so 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 it means we can actually map any point in upper plane to circle center $(0,0)$).Well the function name is rather long,I think you know what it means.Now we will find a way to map the strip (1) to upper plane.First revolve: $f_2(z)=e^{\frac{i\pi }{2}}$,and (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 $f_3(z)=2\pi z$,and it maps strip(2) to strip(3),the strip between 2 lines: $Imz=0*2\pi=0$,$Imz=\frac{1}{2}*2\pi=\pi$...(3)
Step4:Let $f_4(z)=e^z$,we finally map (3) to upper half plane.
Let g(z)=$FROMUPPERTODISC(z)\cdot f_4(z)\cdot f_3(z)\cdot f_2(z)\cdot f_1(z)$
g(z) is the function you want.