A Gallery of Complex Complex Functions

     A tour of this gallery will introduce you to some fascinating graphs of some very complicated complex valued functions.  For a description of the technique used to display these graphs click here:  Color Graphs of Complex Functions    The functions plotted are the solutions, G, to a form of Schroeder's functional equation
G(f(z)) = a G(z)
where f is analytic at zero, f(0) = 0, and f '(0) = a.  (G is actually the inverse of the clasical Schroeder function for f.) 
     For the functions plotted here, f was chosen to be a ratio of quadratic polynomials with |a| > 1.  The functions G have been approximated using the limit
G(z) = lim f n(z/a n)
where f n denotes the nth iterate of  f.  All of the functions, G, determined in this way are meromorphic.  That is, in any compact subset of the complex plane, G is analytic except for finitely many poles.  As a measure of the complexity of these functions it is interesting to note that for many of the functions plotted here, the number of poles in the plotted region easily exceeds one million, and for several probably exceeds one billion. 
     One of the interesting aspects of the graph of G(z) is that the colors of the almost constant regions correspond to the attracting fixed points and attracting fixed point cycles of f(z).  Specific comments will be made for some of the graphs.  There is an extraordinary variability in the graphs of G which can be obtained by changing f.  This gallery shows some of my favorites found in about ten years of exploring.
Larry Crone Home Page
American University Department of Mathematics and Statistics