Domain Coloring

Domain Coloring

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Domain Coloring

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Developer’s Description

By VoiceSync
With this application you can select a predefined formula or create your own one, you can also modify at your convenience a predefined one, a multiple 3d object animation or fixed image panel is displayed with generated color domain. Usage is simple and intuitive: * Select a predefined formula from the list or * Modify any formula or build a new one, use ‘z’ as te complex variable, press to evaluate * Predefined functions are: sin, cos, tan , exp, log, acos, asin, atan * Operators: () * / – + ^ * examples: z 1/z * sin(1/z) log(z)/z * click on the animation panel to switch between fixed image/animation * images can be saved in PNG loos less format at any resolution (render time may be long)
Representation of complex functions with the method of domain coloring. The online version of these applets is slow. I recommend to download all the files for better performance.

Domain Coloring Editor’s review

FromVoiceSync:Domain coloring is a technique for visualizing functions of a complex variable. You can select one of the preset functions or create/modify your own, rendering is performed combining multi threading so all available cores of CPU are used and a fast zCompiler to achieve near real time display. Results can be exported to a .png or .jpg file in selectable resolution up to 32k width

Code generates HSV color representation of complex function.
domainColoring.m : this MATLAB function returns different representations of complex function.
veryComplex.m : contains collection various complex functions and their visualization using domainColoring.m.
testDomainColoring.m : simple script that demonstrates the use of domainColoring.m

Reference:

Domain coloring (cf. section 5.2) is a technique that makes use of a color scheme to visualize complex valued functions. The left image shows a reference color scheme that represents the complex plane. The right image shows a typical color plot of a meromorphic function. The black spots denote zeros, here with multiplicity 2 (the bigger one on the left) and 1 (the one on the right). The white spot in the upper third is a double pole whereas the punctual spot at the bottom is a simple pole. This can also be seen from the multiplicity and order of the colors around these points – the double zeros and poles are surrounded by the complete color circle twice in contrast to the simple zeros and poles.

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in \({\mathbb R}^3\) from explicitly given branch points with corresponding branch…
Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in \({\mathbb R}^3\) from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical projection onto \({\mathbb R}^2\) and can therefore be considered as multivalued graphs over the plane – hence they provide a comprehensible visualization of the topological structure. Complex functions are elegantly visualized using domain coloring on a subset of \({\mathbb C}\). By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.

A domain coloring or phase portrait is a popular and attractive way to visualize functions of a complex variable. The absolute value (modulus) of the function at a point is represented by brightness (dark for small modulus, light for large modulus), while the argument (phase) is represented by hue (red for positive real values, yellow-green for positive imaginary values, etc.). This Demonstration creates domain colorings for many well-known functions in complex analysis: trigonometric and exponential functions, Möbius transformations, the Joukowski map, the Koebe function, Weierstrass  functions, gamma and beta functions, and the Riemann zeta function.

DETAILS

The terms domain coloring or phase portrait refer to the type of visualization technique for functions of a complex variable used here, or some variation thereof. For a function  of a complex variable, each point  in the complex plane is colored according to the value of . The brightness represents the absolute value (or modulus) of , with light representing large absolute value and dark representing small absolute value. The hue represents the argument (or phase) of , with red representing positive real numbers, yellow-green representing positive imaginary numbers, cyan representing negative real numbers, and so forth.

In addition to the standard polynomial, trigonometric, and exponential functions, we have included: the Möbius transformation , the Joukowski map , the Koebe function , a few instances of the beta functions , the gamma functions  and , and the Weierstrass ℘ functions  defined by  if , and finally the famous Riemann zeta function , defined as the analytic continuation of the Dirichlet series  for .

-Domain colouring techniques in GeoGebra in order to obtain maps of complex function with complex variable and in the Riemannian sphere. More Recently, GeoGebra can reproduce the domain colouring techniques in 2d and 3d Windows (see fig. 4). These techniques in GeoGebra were used for the first time by Ana Breda and José Dos Santos to study the maps of function of ℝ 2 in ℝ 2 and maps of complex function with complex variable (Breda, A., Dos Santos, J. & Trocado, A., 2013); moreover, they were applied to represent maps in the Riemannian sphere and study the Möbius transformations in the Riemann sphere with GeoGebra (Breda, A. & Dos Santos, J., 2015). The colouring domain applied in GeoGebra provides a way to visualise graphs in four dimensions and extend the field of application of this software to research and educational purposes.

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