Complex Analysis is one of the venerable pillars on which modern mathematics rests, and to which such diverse fields as Algebraic Topology and Algebraic Geometry owe their origins. And while it enjoys a long and distinguished tradition that parallels the analysis of real functions, analytic maps also have some deep connections with geometry that are somewhat distinct from those of Real Analysis. The condition of differentiability for maps from

has an interesting geometric interpretation (the Cauchy-Riemann equations) that leads to such beautiful novelties as their conformality away from singularities, Cauchy's Integral Theorem, the Riemann mapping theorem, and Lucas' Theorem on the disposition of the zeros of the derivative of a polynomial in relation to the disposition of the zeros of the original function. Marden's theorem, which we illustrate and prove in this Microworld, is a surprising special case of Lucas' Theorem. But maps from the plane to the plane are in general difficult to visualize, simply because they require four dimensions (degrees of freedom) to represent them.

This 3D Mathwright Microworld aims to give you the opportunity to investigate some of these things for yourself. Along the way, you can make your own gallery of color portraits. It is a 3D Microworld that uses orthographic rather than perspective projection. Everything happens in a plane, so there is no need for the third dimension (Four would be nice, alas).

One of the great strengths of 3D Graphics is its versatile use of color and light. Imagine that each point in the complex plane is represented in polar coordinates: where is the distance from the origin or the modulus, and is the angle that the positive real axis makes with a directed ray from the origin to the point. This angle is called the argument of the complex number. If we let represent intensity and represent hue as an additive combination of the three basic colors, Red, Green, and Blue, then each point in the plane is uniquely represented by a color with black at the origin. While such a choice is rather arbitrary, once we establish a correspondence between complex numbers and colors, we will have a bridge to the visualization of these maps. The following is one such association:

 

Given this correspondence, any function mapping the complex plane to itself may be represented by a picture in which each domain point is given the color of its image. It is probably easy to understand the rotation given by multiplication by .

 For example, the polynomial function has the following picture:

 

It has 6 easily discernible zeros, the sixth roots of unity, and those zeros interact with one another in interesting and colorful ways. For example, the center of the picture is dominated with function values that are close to negative reals, as one would expect. Since is a primitive sixth root of unity, we see the "snowflake" or hexagonal symmetry of the map under rotations through and its multiples.

Another fact, easily portrayed by the picture, is the simple nature of the zeros. Around each zero, the function makes a single circuit of the origin. Each zero has order 1, and this implies that the function is locally invertible in the vicinity of each zero. It is reflected in the fact that there is around each zero, a single circuit of the colors from red to yellow to green to blue and violet and back to red.

If a complex mapping has only zeros and poles of finite order, then for any simple closed differentiable curve in the plane that does not pass through any zeros or poles of the complex mapping, we can ask what number of zeros and poles, counted algebraically with their multiplicities, are contained within the interior of the curve. Visually, it is the integer number of times the curve is "wrapped" around the origin by the complex mapping. We see that number easily with a color portrait, simply by reading how many times the color spectrum

is traversed in one counterclockwise traversal of the curve. A pole gives color traversal in the opposite sense. This integer 'winding number' is the algebraic sum of the number of zeros and poles, counted with their multiplicities, that are contained within the curve. When there are multiplicities, small perturbation of the function yields bifurcation where only simple zeros and poles close to the original ones are found.

 

The Microworld features a new capability of Mathwright (available since version 2.13, Sept 14, 2003) that makes use of 3D graphics and the built-in complex number type to draw fast color portraits of complex maps, thereby giving the reader the chance to experiment and to explore the properties of complex functions at her own pace, and with her own questions.