Date: Oct 7, 2013 8:19 AM
Author: David Park
Subject: Re: plotting complex functions in (x,y,t) space

Michael,

It is rather difficult to understand what the form of your function is and
what the role of t is. Not understanding that, it is even more difficult to
understand why you are trying to plot it as you describe. Maybe you could
clarify the question.

I sell (for $50) a Mathematica Application, Presentations, which has rather
extensive facilities for plotting complex functions, mostly directly in
terms of a complex variable z. Among other things it has DomainColoring,
which is something like what you describe, along with various coloring
functions and the ability to specify your own coloring. For drawing paths in
the complex plane it has ComplexCurve (suggested by Murray Eisenberg) so,
for example, you could draw a logarithmic spiral by ComplexCurve[Exp[(0.1 +
3 I) t], {t, -12, 6}]. It also has the ability to handle finite numbered
multifunctions.

My feeling is that domain coloring at first sounds great, and makes great
art work, but is not so good at extracting precise information about a
complex function. For one thing, people do not have an intuitive feel for
the coloring functions - whatever they are. Multiple presentations with
dynamic numerical values is usually better. One method that I like is to
drag a single locator around the complex plane, perhaps with a background
showing modulus, and attach an arrow to the locator representing the complex
value at that point. In a sense this is a local 4-D display because the
plane provides two dimensions and the arrow provides two more. If you
combine this with dynamic display of the numerical values you can explore
the space and extract precise information. It even works for a
multi-function where you can smoothly transition from one branch to another.


David Park
djmpark@comcast.net
http://home.comcast.net/~djmpark/index.html



From: Michael B. Heaney [mailto:mheaney@alum.mit.edu]


Hi,

I'd like to plot, on [x,y,t] axes, a complex function F[x,y,t], with the
magnitude of F represented by opacity, and the phase of F represented by
color. Does anyone have suggestions on how best to do this?

Thanks,

Michael