Suggestion: Visualization of complex functions with Mathematica
"René Böhlendorf"
I am an owner of Mathematica 4.2.0.0 and got
my diploma in mathematics from Technische Universit=E4t Berlin in the
year 1981. My thesis was about complex analysis with the title 'Der
universelle Teichm=FCllerraum', which is a space of conformal mappings
from a part of the complex sphere with quasiconformal continuations to
the complete complex sphere.
I was always interested to visualize complex functions, but the possi-
bilities are rather limited as we can visually imagine only threedimen-
sional objects and the graph of a complex function of one variable
would be a fourdimensional object.
I was wondering if your program Mathematica would try to offer some-
thing to visualize complex functions in terms of four dimensions, but
I did not find something which is quite understandable.
Nevertheless I want to offer two ideas to you which you could evaluate
and possibly implement in future versions of Mathematica. I understand
that Mathematica aims mainly at teaching mathematics besides being a
general help for technicians, physicists, mathematicians and all other
persons applying mathematics.
Visualizations of complex functions are up to now only graphs of the
absolute value, the real or the imaginary part of the function. This
gives only a poor impression similar to seeing only a shadow of an
object instead of the object itself.
1. My first idea is to view the complex plane as a normal twodimensio-
nal Euclidian plane and attach a twodimensional vector to each point
to represent the value of a complex function there.
Of course there are many problems to view such an object. It would
look more or less like a cornfield and it would be difficult to get
a good impression of the behaviour of the complex function which is
displayed.
I thought about drawing such a picture with help of Mathematica or
other programs for some simple functions and use only a small set of
points, but found it too troublesome for me and of no general value.
A function in Mathematica which would draw such a 'cornfield' auto-
matically using a raster would save me this trouble and be a great
fun for me! Although it would be not too interesting for professio-
nal mathematicians who work in complex analysis it could serve as a
good help to teach complex analysis for beginners by showing true
graphs of a complex function.
2. My second idea is to use colour and it is perhaps the better one.
The complex plane or sphere would be coloured in such a way that
different regions have different colours, perhaps even using con-
tinuously changing colours.
The visualization of the complex function would be a second plane
or sphere showing the same colours as the first one, but moved to
the place they get by applying the complex function. I believe that
this will produce good and impressive pictures for many complex
functions although I did never see such pictures up to now!
As I worked in the area of software development since 1981 I am able
to estimate that quite a lot of work would have to be done if you would
decide to implement one or both of my ideas in Mathematica.
At least I hope that you will think the ideas over even if you should
decide not to implement them. A few lines as a response to me when it
is convenient for you would make me happy!
Faithfully yours,
Ren=E9 B=F6hlendorf
Steve Luttrell
Add-ons/StandardPackages/Graphics/ComplexMap entry in the Help browser.
--
Steve Luttrell
West Malvern, UK
""René Böhlendorf"" <R.Boeh...@t-online.de> wrote in message
news:bqkb3v$hkt$1...@smc.vnet.net...
Arturas Acus
as far as concerns your second suggestion, you can
take a look at this book:
http://www.amazon.com/exec/obidos/tg/detail/-/0387989293/103-0551293-2771067?v=glance
It deals with quantum wave function visualization (you know they are
complex functions) concept. The author provides background on color
models, etc... all pictures are made with Mathematica. Very impressive pictures.
Peltio
>I was always interested to visualize complex functions,
Have a look at this website
www.kfunigratz.ac.at/imawww/thaller
where Bernd Thaller has put some pages illustrating his ComplexPlot.m
package.
The package was described in The Mathematica Journal, too, but I don't
remember which issue it was.
At any rate the package, and further material, was downloadable from
Thaller's website.
www.kfunigratz.ac.at/imawww/thaller/visualization/complex.html
Check it out to see if it is still so.
cheers,
Peltio
--
Invalid address in reply-to. Crafty demunging required to mail me.
Murray Eisenberg
is not being found.
Peltio wrote:
> René Böhlendorf wrote:
>
>
>>I was always interested to visualize complex functions,
>
>
> Have a look at this website
> www.kfunigratz.ac.at/imawww/thaller
> where Bernd Thaller has put some pages illustrating his ComplexPlot.m
> package.
> The package was described in The Mathematica Journal, too, but I don't
> remember which issue it was.
> At any rate the package, and further material, was downloadable from
> Thaller's website.
> www.kfunigratz.ac.at/imawww/thaller/visualization/complex.html
>
> Check it out to see if it is still so.
>
> cheers,
> Peltio
> --
> Invalid address in reply-to. Crafty demunging required to mail me.
>
>
--
Murray Eisenberg mur...@math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
Peltio
>What is the correct URLs for the ones listed below: www.kfunigratz.ac.at
>is not being found.
I apologize.
The correct URL (I checked it out now) is
http://www.kfunigraz.ac.at/imawww/vqm/pages/dlgraph.html
While searching the web for the correct address, I've also found the
following packages that can be used to plot complex functions (I'm sure that
there are many more). Some are new to me, but the original poster might find
them interesting enough to try'em out.
They can be reached by using the author name, the package name in Google
(www.google.com) For example fill in the search field with "ComplexPlot.m"
or "complex mathematica filetype:m".
Complex.m by Goeran Andersson
http://www.funet.fi/pub/sci/math/mathematica/Analysis/Complex.m
*zListPlot[ zlist, opts] pointplots the complex numbers in zlist.
*zwPlot[ z, w, {t, tmin, tmax}, opts] plots the complex mapping
w=f(z), z=z(t).
ComplexPlot.m by Jeff Olson
http://www.ph.utexas.edu/~jdolson/math/ComplexPlot.m
*ComplexPlot[func, {fx, fy}, {t, tmin, tmax}]
*ComplexListPlot[{z1, z2, ...}] plots a list of complex numbers.
*ComplexVectorPlot[func, {xmin, xmax}, {ymin, ymax}]
*ComplexPartialPlots[func, {xmin, xmax}, {ymin, ymax}]
*ComplexColorPlot[func, {xmin, xmax}, {ymin, ymax}]
*ComplexContourPlot[func, {xmin, xmax}, {ymin, ymax}]
*CartesianMap[f, {x0, x1, (dx)}, {y0, y1, (dy)}] plots the image
of the cartesian coordinate lines under the function f.
*PolarMap[f, {r0:0, r1, (dr)}, {phi0, phi1, (dphi)}] plots the
image of the polar coordinate lines under the function f.
ComplexPlot3D by Kevin McIsaac
*ComplexPlot3D[fn, {x, x0, x2, (dx)}, {x, x0, x2, (dx)}, (options)]
Plots in the complex plane. The absolute value is represented
by the height and the phase is represented by the color
ComplexMap by Roman Maeder
*CartesianMap[f, {x0, x1, (dx)}, {y0, y1, (dy)}] plots the image
of the cartesian coordinate lines under the function f.
*PolarMap[f, {r0:0, r1, (dr)}, {phi0, phi1, (dphi)}] plots the
image of the polar coordinate lines under the function f.
Transform2DPlot by Xah Lee
http://www.xahlee.org/SpecialPlaneCurves_dir/MmaPackages_dir/Transform2DPlot
_dir/Transform2DPlot.m
This package exports the function Transform2DPlot and
Transform2DGraphics that plot the image of the plane under
arbitrary transformation function f:R^2->R^2 or f:C^1->C^1.
It can be easily adapted to make it plot conformal maps.
It is possibile to write a shell that calls the procedures with a
syntax similar to that of ComplexMap. (years ago I wrote one,
only a few lines long - no big deal since Xah Lee's procedures do
it all. If someone is interested, though, and has already
downloaded the original package by Xah Lee let me know and
I'll send it to them)
ComplexMapPlot by Theodore W. Gray and Jerry Glynn
This one is included in their book "Exploring Mathematics with
Mathematica", Addison-Wesley, 1991. I don't think it is
available on the web. I had found a reference to it in a gallery
of complex functions (I saved it in rtf format without the
pictures!!! How smart : )). But if the original poster has access to
the book he can find out if what it does suits him.
cheers,
Peltio
David Park
This package contains complex analysis routines and complex graphics
routines. There are routines that convert the regular 2D Graphics into
equivalent complex forms. For example ComplexLine[{z1,z2,z3...}] takes
complex numbers for the point coordinates. There are routines for producing
one or two panel plots or animations of complex functions. Each panel may be
one of the following plot types.
1) Cartesian/PolarSurface - Plots the surface s[f[z]] where f is a complex
function and s is a real function.
2) Cartesian/PolarCoded3D - Plots the surface s[f[z]] and then puts a
colored contour plot on the surface representing r[f[z]] where s and r are
two real functions.
3) Cartesian/PolarGrid - Basically the ComplexMap routines from Mathematica,
but you can combine multiple grids of different colors.
4) Cartesian/PolarContour - 2D colored contour plots with labeling of all or
selected contours.
5) CodedDensity - A density plot using color to code the complex
information. There is one color function provided that codes modulus and
argument. The user can also write and use his own color routines.
6) ComplexVector - Attaches a scaled vector representing the complex value
to each of a set of points. The user supplies whatever set of points he
wishes. This is also useful with animation on a selected point or points to
trace the behavior of the function on paths in the complex plane.
7) RiemannSphere - Maps any set of graphics primitives, but usually a
complex map grid, through a complex function to the Riemann sphere.
The animations are rotation of 3D surfaces and homotopies between two
complex functions. Other custom animations may also be produced. All of the
plot types also allow the user to add extra graphics primitives to embellish
plots
David Park
dj...@earthlink.net
http://home.earthlink.net/~djmp/
Murray Eisenberg
http://home.earthlink.net/~djmp/Mathematica.html
I've been using it in an undergraduate complex analysis course this
semester and have found it VERY instructive.
In case anybody wants to examples beyond those in Park's notebooks, you
may take a look at some of the notebooks at
http://www.math.umass.edu/Courses/Math_421/Files/files.html
namely, the following files:
IntroComplexGraphics.nb
CartesianPolarForms.nb
VisualizeFunctions.nb
ComplexCurves.nb
VisualizeFunctions2.nb
RiemannSphere.nb
ExponentialFunction.nb
Sine.nb
Poles.nb
Peltio wrote:
> "Murray Eisenberg" wrote
>
>
>>What is the correct URLs for the ones listed below: www.kfunigratz.ac.at
>>is not being found.
>
>