Results 1 to 2 of 2

Math Help - Complex Analysis Linear Fractional Transformation

  1. #1
    Newbie LeraysJeans's Avatar
    Joined
    Dec 2011
    Posts
    7

    Complex Analysis Linear Fractional Transformation

    I am stuck on a LFT question,
    T maps the real axis onto itself and the imaginary axis onto the circle  \mid w-\frac{\1}{2} \mid  =  \frac{\1}{2}
    I do not have a good grasp on LFTs, although I understand fixed points and triples to triples techniques and can solve problems like,
    Find a LFT that carries the circle  \mid z \mid = 1 onto the line  Re((1+i)w)=0
    But when I have to take into account multiple properties of transformation I get lost, I've read my text a bunch and even picked up another, not to mention the countless number of help articles I've read online. I'm not sure what to do. I've attempted to use fixed points and triples to triples techniques to no useful end. Any hints or reference material would be greatly appreciated!
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie LeraysJeans's Avatar
    Joined
    Dec 2011
    Posts
    7

    Re: Complex Analysis Linear Fractional Transformation

    So I think I solved this one,

     T(z)=\frac{z-\infty}{z}\frac{i}{i-\infty}

     T(z)=\frac{zi-\infty i}{zi-z\infty}

    yada...yada... do the same triplet to triplet thing for the other set of points... and its inverse is

     S^-^1(w)= \frac{z(\frac{1}{2}+\frac{i}{2})}{z(\frac{1}{2} + \frac{i}{2})+(\frac{1}{2} - \frac{i}{2})}

    S^-^1(T(z)) =  \frac{\frac{zi-\infty i}{zi-z\infty}(\frac{1}{2}+\frac{i}{2})}{\frac{zi-\infty i}{zi-z\infty}((\frac{1}{2} + \frac{i}{2})+(\frac{1}{2} - \frac{i}{2}))}


     S^-^1(T(z)) = \frac{(zi-\infty i)(\frac{1}{2}+\frac{i}{2})}{(zi-\infty i)(\frac{1}{2} + \frac{i}{2})+(zi-z\infty)(\frac{1}{2} - \frac{i}{2})}

    take the limit...

     L(z)=\frac{\frac{1}{2}-\frac{i}{2}}{\frac{1}{2}-\frac{i}{2}-\frac{z}{2}+\frac{zi}{2}}

     L(z)=\frac{1}{1-z}

    And it turns out that this transformation leaves the real axis alone... this was a little planing (with choosing the triples) but mostly luck... Does anyone know a better way of solving this?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Möbius transformation and fixed points - complex analysis
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 13th 2010, 10:52 AM
  2. Replies: 3
    Last Post: September 12th 2010, 07:35 PM
  3. Complex analysis, find the image of a transformation
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: July 14th 2010, 10:54 AM
  4. Question on linear transformation in analysis
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 10th 2009, 08:07 AM
  5. complex analysis: linear transformations
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 16th 2009, 02:38 PM

Search Tags


/mathhelpforum @mathhelpforum