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Visual Complex Functions: An Introduction with Phase Portraits

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The graph of an analytic function f(z) naturally lives in two complex dimensions or four real dimensions. Therefore we cannot visualize such graphs directly. There are several ways to work around this limitation.

One way is to use before-and-after plots: here’s a region of the complex plane, and here is its image under f. Or we could graph |f|, the absolute value of the function. Another way to visualize analytic functions is to view the range in (r, θ) polar coordinates and encode the value of θ as a position on a color wheel. A pure phase plot would graph the θ component of f(z) as a function of z.

There are several variations on the phase approach that add information regarding the r component of f(z). One would be to create a 3-D plot using height to indicate |f(z)| and color to indicate the phase. If we want to stay in two dimensions, we could add contour lines for |f(z)| or map the θ component to hue and the r component to saturation in a HSV (hue, separation, value) description of color.

All the methods above are used in Visual Complex Functions by Elias Wegert, though pure phase plots are used most often. True to its title, Visual Complex Functions emphasizes visualization. However, it is not simply a book on visualization. It is an introductory complex analysis book with an unusually heavy emphasis on visualization. The standard topics — power series, residues, the Riemann mapping theorem, etc. — are all included.

One drawback to phase plots is that the mapping of phase to color is subjective. However, the qualitative information apparent in a graph does not rely of the specific mapping. You can, for example, spot zeros of a function by the colors rotating one way and poles by the colors moving in the opposite sequence. You can tell the degree of a zero or pole by how many times the colors cycle around the point. And you can spot an essential singularity by a flurry of change.

The emphasis on visualization gives the reader a deeper intuition for the behavior of complex functions. Experienced mathematicians may be surprised by the new insight this approach gives. The author explains his own excitement regarding visualization as follows.

My acquaintance with complex functions dates back almost forty years, but it took a long time until I could begin to see my friends. I love them even more ever since I know their phases. This book has been written to let you share my joy.

Visual Complex Functions contains over 200 images, many of these quite stunning. The image below is a phase plot of one of the functions plotted in the book. However, the book did not specify the scale and so the image below, while equally interesting, does not reproduce the image in the book.

See the paper Phase Plots of Complex Functions: A Journey in Illustration for a more detailed summary of the book. Visual Complex Functions is in part a book-length elaboration of this paper.

Visual Complex Functions is the first volume of a series. Wegert says that the second volume “will be devoted to selected topics and various applications of complex methods, like integral transforms, boundary value problems, and signal analysis.”

John D. Cook is an independent consultant and blogs at The Endeavour.

Date Received: 
Thursday, February 21, 2013
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Elias Wegert
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John D. Cook
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Saturday, March 16, 2013