Suggestion: Visualization of complex functions with Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg44837] Suggestion: Visualization of complex functions with Mathematica
- From: R.Boehlendorf at t-online.de ("René Böhlendorf")
- Date: Wed, 3 Dec 2003 04:24:24 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Dear Madams and Sirs, 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
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