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Images in Complex Analysis

  1. Nov 13, 2007 #1
    1. The problem statement, all variables and given/known data

    f(z) = (z+1)/(z-1)
    What are the images of the x and y axes under f? At what angle do the images intersect?

    2. Relevant equations

    z = x + iy

    3. The attempt at a solution

    This is actually a 4 part question and this is the part I don't understand at all really.
    The first 2 parts were a) Where is f analytic? Compute f' for this domain. and b) Where is f conformal.

    I concluded that f is not analytic because it isn't differentiable at z = 1. The derivative, d/dz = z/(z-1) - (z+1)/[(z-1)^2]. I said f' is conformal along the complex plane except where z = 1 as well. z=1 creates problems in the derivative, where you measure what is and isn't conformal and where. I'm not sure this information is relevant to the actual images though, but I thought I'd put it in anyway just in case.

    Thanks.
     
  2. jcsd
  3. Nov 13, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    f is analytic, except at z=1. It's just not holomorphic. The x and y axes intersect at z=0. That's not a point where z=1 creates a problem. And f is conformal everywhere it's analytic. Once you've actually computed the images of the axes you can confirm that it's conformal.
     
  4. Nov 13, 2007 #3
    That information is indeed relevant, especially the fact the mapping will be a conformal mapping. What do you know about conformal maps? Why will this be important when we're say...computing the angles that the axes intersect?

    Choose a few points on the axes, say -1, 0, 1, i, -i, and find their images under the mapping. And technically shouldn't that be the real and imaginary axes rather than the x and y axes?

    In general, this mapping will send planes and circles to planes and circles, if that helps at all.
     
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