1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Some examples of Möbius transformation

  1. Jun 5, 2014 #1
    1. The problem statement, all variables and given/known data
    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##


    2. Relevant equations

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


    3. 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?
     
  2. jcsd
  3. Jun 6, 2014 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Some examples of Möbius transformation
Loading...