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Math Help - Mobiust transformations

  1. #1
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    Mobiust transformations

    Find a Mobius Transformation that takes the circle modulus z=1 to the straight line x+y=1.
    I know I need to pick 3 points, but I don't know what to do then. Like if I pick out the point (0,0) or (0,i) do I plug these values into x+y=1. I'm just really confused on this step.
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  2. #2
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    Quote Originally Posted by kathrynmath View Post
    Find a Mobius Transformation that takes the circle modulus z=1 to the straight line x+y=1.
    I know I need to pick 3 points, but I don't know what to do then. Like if I pick out the point (0,0) or (0,i) do I plug these values into x+y=1. I'm just really confused on this step.
    Here is an easier way to do it. The function f_1(z) = (1-z)/(1+z) takes the circle |z| = 1 to the line x=0. The function f_2(z) = e^{i\pi/4}z takes the line x=0 to y=-x. The function f_3(z) = z+1 takes the line y=-x to y=1-x, which is the desired line. Thus, f_3\circ f_2\circ f_1(z) is a Mobius transformation which does this mapping.

    Here is an interesting thread on this topic.
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