Complex Function Plots

Welcome to the colorful world of complex function graphs

Each of the images shown here is the graph of a function of a complex variable.  The complex numbers result from mathematicians insistence on combining an "imaginary" square root of negative one with the "real" numbers we know and love.  The marriage of imaginary and real numbers provides a framework in which more than a few awkward facts about real numbers become smooth and simple.  For example, in the world of complex numbers every nth degree polynomial can be written as a product of n linear factors. In the "real" world, every nth degree polynomial can be written as a product of linear and irreducible quadratic factors.   It is impressive that perhaps the simplest proof of the factorability of real polynomials follows from the complex version.  The complex number system has very practical results and has become invaluable to science and engineering.  This is one example of the beauty inherent in the abstract side of mathematics.  

Plotting a graph of a real valued function of a real number is relatively simple since only two real numbers are involved.  The graph of a complex valued function of a complex number involves four real numbers and that presents serious challenges.  The availability of color monitors and color printing makes possible a new technique: using color to represent the value of the complex valued function.  The images on the carousel above are graphs of very simple complex functions using such a method.  The images flanking the carousel and those below are graphs of significantly more complicated functions.  The remainder of this site will introduce you to a class of functions impressive for their mathematical significance and the beauty of their graphs. 

About the author:

Larry started working with color graphics in the early 1980s with data from the Coastal Zone Color Scanner (CZCS) for National Oceanic and Atmospheric Administration (NOAA) and soon realized the possibility of using color to plot complex valued functions.  He was overwhelmed by the beauty and complexity of graphs of the Schroeder function and its inverse and has been trying to spread the word since then.  Unfortunately that was a time when no mathematics journals published in color and personal computers had minimal color capability if any.  

He has two Visual Basic programs, Gplot and fPlot, which are available for download from his American University site.  These are 2014 vintage and Larry, who is vintage 1941, would be happy to talk to anyone of perhaps more recent vintage who might be interested in getting involved with the project.  Gplot is the program which produced the bulk of the images on this site.  fPlot is a general purpose complex function plotter which takes a formula and plots the graph.   (For example, plotting a polynomial makes the locations of roots visually obvious.)  The images on the carousel above were produced by fPlot.

Contact: LCrone@american.edu