Conformal map to convert circle to a line

[shehry1]

I need a conformal mapping that would map an ellipse or a circle to a line. I need this for the http://physics.indiana.edu/~berger/p506_fall2008/p506ps6.pdf" .

As far as I can understand, z^2 + 1/z^2 makes the geometry similar to that of a plane on the horizontal axis with a circle centered on the origin. I can use the method of images (I think) but I was thinking of another map that would flatten out the circle and keep that plane in its current form.

Not that it matters, I tried to tailor the homographic projection to my needs but couldn't get it to work.

 

[Ben Niehoff]

z^2 + 1/z^2 does flatten out the circle, if the circle has radius 1. You need to modify this map to flatten out a circle of radius a.

It is easiest to think about what the map does to each of the three boundary surfaces, independently; i.e., what happens to points on the real axis > a, on the imaginary axis > ia, and on the quarter-circle of radius a. If you can't visualize what happens, then try plugging in a complex number into the map to see where it goes (i.e., try x, iy, and ae^(it)).

 

[shehry1]

z^2 + 1/z^2 does flatten out the circle, if the circle has radius 1. You need to modify this map to flatten out a circle of radius a.

It is easiest to think about what the map does to each of the three boundary surfaces, independently; i.e., what happens to points on the real axis > a, on the imaginary axis > ia, and on the quarter-circle of radius a. If you can't visualize what happens, then try plugging in a complex number into the map to see where it goes (i.e., try x, iy, and ae^(it)).

Hmmm...seems very simple now that you have put it that way - guess that's the whole thing. Thanks a lot. Would try it as soon as I get the time

 

[Hurkyl]

Also, keep in mind a line is just a circle passing throgh infinity. (In this geometry, I mean)