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How coordinate lines transform under ##e^z=\frac{a-w}{a+w}##

  1. Sep 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Say how coordinate lines of the z plane transform when applied the following transformation

    ##e^z=\frac{a-w}{a+w}##

    2. Relevant equations


    3. The attempt at a solution
    This is exactly the way the problem is stated. It is a pretty weird transformation in my opinion and I'm guessing ##w## is the transformation and the coordinate lines in question are cartesian coordinate lines.

    That said I've solved for ##w##

    ##w=\frac{a(1-e^z)}{1+e^z}##

    And what I've attempted to do is consider the cases ##x=C## ##y=C## where ##C## is some arbitrary constant and find an expression of the form ##w=u+iv##. This has proven quite tedious and very little if at all illuminating.

    So I was wondering how I could proceed?

    At first I thought this was similar to a bilinear transformation but now I've thought not at all.

    I don't know this topic too well so I'm probably not seeing something basic here, or failing to identify the class of transformation in question


    Thanks.
     
  2. jcsd
  3. Sep 13, 2015 #2

    andrewkirk

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    I suggest you start with discovering how the x axis (real axis) transforms. Calculate w for x values of ##-\infty,-1,0,+1,+\infty##. You'll find a nice simple result.

    Next look at how coordinate lines parallel to the x axis transform. They will be of the form ##a\frac{1-e^{ni+x}}{1+e^{ni+x}}
    =a\frac{1-e^{ni}e^x}{1+e^{ni}e^x}## for ##n## an integer. Again calculate w for x values of ##-\infty,-1,0,+1,+\infty##.

    I think it gets more complicated after the first step. A quicker and easier way to get a feel for it (that would not be available in an exam, so be cautious if this is exam practice) would be to use a spreadsheet or R/matlab-type program to calculate w for the table of coordinate grid points in the square [-10,10] x [-10,10]. I expect a pattern will emerge.
     
  4. Sep 13, 2015 #3
    I just tried plotting the constant x surface on matlab and got an oval but strangely its contour is fuzzy (by which I mean its not a perfect line)
     
  5. Sep 13, 2015 #4
    The oval might be a circle I'm adjusting the scale...

    edit: Indeed it is a circle.
     
  6. Sep 13, 2015 #5
    It seems the constant x and y surfaces map into circles and rays. I dont see why...:nb) Any ideas?
     
  7. Sep 13, 2015 #6
    Nevermind they seem to map into some weird collection of circles
     
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