Images in Complex Analysis

  • Thread starter desaila
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Homework Statement



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?

Homework Equations



z = x + iy

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.
 

Answers and Replies

  • #2
Dick
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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.
 
  • #3
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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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