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| [[Image:Color complex plot.jpg|200px|right|thumb|Domain coloring plot of the function
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| <br>ƒ(''x'') =(''x''<sup>2</sup> − 1)(''x'' − 2 − ''i'')<sup>2</sup>/(''x''<sup>2</sup> + 2 + 2''i''). The hue represents the function argument, while the saturation and value represent the multiply-wrapped magnitude.]]
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| '''Domain coloring''' is a technique for visualizing functions of a [[complex variable]]. The term "domain coloring" was coined by Frank Farris possibly around 1998.<ref>Frank A. Farris, [http://www.maa.org/pubs/amm_complements/complex.html Visualizing complex-valued functions in the plane]</ref><ref name="Ludmark1">{{Cite web|url=http://www.mai.liu.se/~halun/complex/domain_coloring-unicode.html|title=Visualizing complex analytic functions using domain coloring|accessdate=2006-05-25|year=2004|author=Hans Lundmark}} Ludmark refers to Farris' coining the term "domain coloring" in this 2004 article.</ref> There were many earlier uses of color to visualize complex functions, typically mapping argument (phase) to hue.<ref>
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| {{cite journal
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| | journal = Pixel: the magazine of scientific visualization
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| | title = A Color Gallery of Complex Functions
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| | author = David A. Rabenhorst
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| | volume = 1
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| | issue = 4
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| | publisher = Pixel Communications
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| | pages = 42 et seq
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| | issn =
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| | date = 1990
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| | url = http://books.google.com/books?id=DRUSAQAAMAAJ&q=%22phase%22+%22hue%22+complex-functions&dq=%22phase%22+%22hue%22+complex-functions&hl=en&sa=X&ei=9iCFT4SRL8WdiAKb5KT1BA&ved=0CDMQ6AEwAA
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| }}</ref> The technique of using continuous color to map points from domain to codomain or image plane was used in 1999 by George Abdo and Paul Godfrey<ref name="Abdo1">{{Cite web|url=http://my.fit.edu/~gabdo/|title=Plotting functions of a complex variable: Table of Conformal Mappings Using Continuous Coloring|accessdate=2008-05-17|year=1999|author=George Abdo & Paul Godfrey}}</ref> and colored grids were used in graphics by [[Douglas N. Arnold|Doug Arnold]] that he dates to 1997.<ref name="Arnold1">{{Cite web|url=http://www.ima.umn.edu/~arnold/complex.html|title=Graphics for complex analysis|accessdate=2008-05-17|year=2008|author=Douglas N. Arnold|authorlink=Douglas N. Arnold}}</ref>
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| ==Motivation==
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| ===Insufficient dimensions===
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| A [[real function]] <math>f:\mathbb{R}\rightarrow{}\mathbb{R}</math> (for example <math>f(x)=x^{2}</math>)
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| can be [[graph of a function|graphed]] using two [[Cartesian coordinates]] on a [[Plane (mathematics)|plane]].
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| A graph of a [[complex function]] <math>g:\mathbb{C}\rightarrow{}\mathbb{C}</math> of one
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| [[complex variable]] lives in a space with two complex dimensions. Since the [[complex plane]] itself is two dimensional, a graph of a complex function is an object in four real dimensions. That makes complex functions difficult to visualize in our three dimensional space. One way of depicting
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| [[holomorphic]] functions is with a [[Riemann surface]].
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| ===Visual encoding of complex numbers===
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| Given a complex number <math>z=re^{ i \theta}</math>, the phase (also known as argument)
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| <math>\theta</math> can be represented by a hue, and the modulus <math>r=|z|</math> is
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| represented by either intensity or variations in intensity. The arrangement
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| of hues is arbitrary, but often it follows the [[color wheel]]. Sometimes the phase is represented by a specific gradient rather than hue. | |
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| [[Image:Unit circle domain coloring.png]]
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| ==Example==
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| The following image depicts the [[sine]] function <math>w=\sin(z)</math> from <math>-2\pi</math> to <math>2\pi</math>
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| on the real axis and <math>-1.5</math> to <math>1.5</math> on the imaginary axis.
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| [[Image:Sine.png]]
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| == See also ==
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| *[[Color wheel graphs of complex functions]]
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| * [[Conformal pictures]] where the domain is colored with a picture and not with a fixed [[color wheel]].
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| ==References==
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| <references />
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| ==External links==
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| *[http://w.american.edu/cas/mathstat/lcrone/ComplexPlot.html Color Graphs of Complex Functions]
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| *[http://www.maa.org/pubs/amm_complements/complex.html Visualizing complex-valued functions in the plane.]
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| *[http://www.cs.berkeley.edu/~flab/complex/gallery.html Gallery of Complex Functions]
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| *[http://alessandrorosa.altervista.org/ Complex Mapper] by Alessandro Rosa
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| *[http://www.jedsoft.org/fun/complex/ John Davis software] - [[S-Lang (programming language)|S-Lang]] script for Domain Coloring
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| *[http://devrand.org/show_item.html?item=72&page=Project Open source C and Python domain coloring software]
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| *[http://www.hansfbaier.de/wordpress/computers-and-mathematics/ Enhanced 3D Domain coloring]
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| *[http://www.codeproject.com/Articles/492355/Domain-Coloring-Method-on-GPU Domain Coloring Method on GPU]
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| *[http://complexanalysis.sourceforge.net Java domain coloring software (In development)]
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| *[[MATLAB]] routines [http://www.mathworks.com/matlabcentral/fileexchange/29028]
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| * [http://www.analysis.uni-hannover.de/~gruber/Computing/Conformal/ Python script for GIMP by Michael J. Gruber]
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| {{DEFAULTSORT:Domain Coloring}}
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| [[Category:Complex analysis]]
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