Complex Function Plots

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.