finding the conformal mapping and plotting complex conformal mapping in Matlab?

Discussion in 'MATLAB' started by Mike, Aug 29, 2006.

1. MikeGuest

Greetings everybody,

I am wondering if there is a way to find out the resultant curve for
the following conformal mapping:

w= (1-i *z) / (1- i * x * z)

where z is on the horizontal line a*i -infinity to a*i + infinity, and
"i" is the imaginary unit, "x" is a parameter.

I am guessing that this curve must be a circle, but I want to locate

Also it would be very helpful to visualize the conformal mapping in
Matlab. Any command for that?

Thanks

Mike, Aug 29, 2006

2. Robert IsraelGuest

Good guess. A fractional linear transformation such as this one
maps (straight lines and circles) to (straight lines and circles).
Since a straight line or circle is determined by three points on it,
all you have to do now is substitute three values for z. I might
suggest z=0 and z=infinity as two of the values. You could also
use the fact that (assuming a and x are real) your curve must be
symmetric about the real axis, because changing z from t+ia to
-t+ia takes w to its complex conjugate.

Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Robert Israel, Aug 29, 2006

3. MikeGuest

Yes, but how to obtain the center and the radius?

Mike, Aug 29, 2006
4. Robert IsraelGuest

Did you try following my hint? What points did you get?

Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Robert Israel, Aug 29, 2006
5. MikeGuest

I've found the circle using your hint that the curve must be symmetric
about the real axis. thanks a lot Robert! You are smart.

Mike, Aug 29, 2006