🚀 go-pugleaf

In <1992Nov21.154035.5381@mnemosyne.cs.du.edu> lmitchel@nyx.cs.du.edu (lloyd mitchell) writes:

>In sci.math, a question was asked about taking the derivative of a non-
>analytic complex function.  In particular, the poster was trying to 
>differentiate

>	f(z) = y + ix, where z = x + iy.

>My question is this:  For such functions, is there any such point that
>corresponds to a critical point for analytic functions?  I'm curious
>because I'd like to generate Mandelbrot-type sets for some functions of
>this nature, and I wonder where one would begin iterating.  (For analytic
>functions, Mandelbrot images are generated by beginning the iteration
>at a critical point.)

>Thanks for any info,
>Kerry Mitchell

You can always regard the complex plane as R^2, so you have a mapping from
R^2 to R^2.  The derivative in this context is the Jacobian matrix, and a
"critical point" would be one where the Jacobian has determinant 0.  These
would tend to occur on curves, however, not isolated points.  The dynamics
of arbitrary differentiable maps from R^2 to itself might be quite 
complicated compared to analytic functions, and I don't think there's an
awful lot that can be said about it in general.  In particular, there may be
no good reason to start iterating at any particular point.

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4