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Visualization of Complex Functions

Bernd Thaller
Journal / Anthology

The Mathematica Journal
Year: 1999
Volume: 7
Issue: 2
Page range: 163-180

Functions f (x, y) of two variables with values in the field of complex numbers (e.g., analytic functions) are often considered abstract mathematical objects which are difficult to visualize. Indeed, the graph of such a function would have to be drawn in a four-dimensional space with coordinates (x, y, Re(f), Im(f)), which cannot be easily done on a sheet of paper. Therefore, several aids for visualizing complex functions have been developed. For example, the standard package ComplexMap.m by Roman Maeder illustrates how the function transforms and distorts the complex plane. Another method uses colors for the visualization of complex values. The standard package ArgColors.m specifies colors to describe the argument of complex numbers. This color map is also used in Kevin McIsaac's package ComplexPlot3D.m [McIsaac 1994]. Here we propose a refined color map of the complex plane. The color map is continuous and one-to-one from the complex plane onto the surface of the color manifold. In addition to describing the argument of a complex number by the hue of the color, the color map uses the lightness of the color to represent the absolute value. Thus the color map is suitable for colored density plots of complex functions. The package ComplexPlot. m provides commands for producing intuitive pictures of complex-valued functions, including density plots and surface plots. Since the human eye usually cannot recognize the precise color-values, our method is not suited for a quantitative representation. However, our method is useful for giving a quick impression of the most important qualitative properties of the complex function, such as the position and order of poles and zeros. The method is particularly useful for the visualization of quantum mechanical wave functions [Pauschenwein and Thaller 1996].

*Applied Mathematics > Visualization
*Mathematics > Calculus and Analysis > Complex Analysis
*Wolfram Technology > Programming > 2D Graphics
*Wolfram Technology > Programming > 3D Graphics
*Wolfram Technology > Programming > Color
*Wolfram Technology > Programming > Graphing and Plotting