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Homework Help: Mapping of circle onto circle

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data

    I know that the Mobius transformation:

    [tex]g(z) = \frac{z-z_1}{z-z_3}\frac{z_2-z_3}{z_2-z_1}[/tex]

    maps a circle (with points z_1, z_2, z_3 somewhere on the circumference) onto a line.

    But, i want a general formula for f(z) that maps a circle (z_1,z_2,z_3) ontp a circle (w_1,w_2,w_3)

    2. Relevant equations



    3. The attempt at a solution

    Does anyone have any ideas?

    Thank you in advance
     
  2. jcsd
  3. Feb 10, 2010 #2
    The inverse of g(z) maps a line into a circle; find the general expression for its inverse, select three points in a convinient line and compose the two.
     
  4. Feb 10, 2010 #3
    I found the inverse. It's z= a function in terms of w and z_1, z_2, z_3.

    Why does this help? and what do you mean 'compose the two' ?
     
  5. Feb 10, 2010 #4
    g(z) takes the points (z1,z2,z3) of the circle to (g(z1),g(z2),g(z3)) in a line. Then choose the inverse f(z) such that it takes the points (g(z1),g(z2),g(z3)) in the line, to (w1,w2,w3) in the circle. The composition f(g(z)) will take (z1,z2,z3) in (w1,w2,w3).
     
  6. Feb 10, 2010 #5
    the inverse f(z) such that it takes the points (g(z1),g(z2),g(z3)) in the line, to (w1,w2,w3) in the circle is:

    [tex]f(g(z))=\frac{w-w_1}{w-w_3}\frac{w_2-w_3}{w_2-w_1}[/tex]

    so now i have to combine this with the g(z) is gave in the first post?
    Why?
     
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