Some examples of Möbius transformation

[skrat]

Homework Statement

Find Möbius transformation that maps:
a) circle ##|z+i|=1## into line ##Im(z)=2##
b) circle ##|z-i|=1## into line ##Im(z)=Re(z)##
c) line ##Re(z)=1## into circle ##|z|=2##

Homework Equations

##f(z)=\frac{az+b}{cz+d}##

The Attempt at a Solution

a) Firstly to move the circle into the origin ##f_1=z+i## than to map it into a line ##f_2=\frac{1-z}{z+1}## than rotate it for ##pi/2## with ##f_3=iz## and lastly move it upwards for ##2i## with ##f_4=z+2i##

So ##f=f_4\circ f_3\circ f_2\circ f_1=f_4(f_3(\frac{1-z-i}{z+1+i}))=\frac{1-z-i}{z+1+i}i+2i=\frac{3i+iz-1}{1+z+i}##

Is that ok?

b) To find a,b,c and d I determine that ##f(0)=0## and ##f(2i)=\infty ## and ##f(-1+i)=i## which gives me ##f_2=\frac{-z}{z-2i}##

Finally I have to rotate the line for ##pi/4## therefore the answer should be

##f(z)=\frac{-z}{z-2i}e^{-i\pi /4}##

c) Well, I know that ##\frac{1-z/2}{1+z/2}## maps circle (with radius 2) into right half-plane.

So I guess ##f(z)=\frac{2-z}{2+z}+1=\frac{4}{z+2}##

Now the inverse transformation is also the answer to part c): ##f(z)=\frac{4-2z}{z}##

What do you think?

 

[haruspex]

You can fairly easily check your answers. Just plug in a couple of different values for z. For (a), start with z = 0.
The way I find much easier is more geometric. If a circle passes through the origin then a simple inversion z→1/z will give you a straight line (and v.v.). The line will be orthogonal to the line joining the origin to the centre of the circle. In (a), this immediately gives you a line parallel to the desired one. Just need to multiply by a suitable real factor.