1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.

  2. jcsd
  3. Nov 13, 2007 #2


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook