Complex Power Series—a Vector Field Visualization

Abstract

How can we visualize the behavior of complex power series and convey to students a clear understanding of convergence and divergence? In this note we describe a visualization technique based on the use of Polya vector fields, as introduced by Polya and Latta [3] and further developed by Braden [1], [2]. In [3], Polya introduced the notion of a vector field representation of a complex function of a complex variable: If f(x + iy) = u(x, y) + iv(x, y), then the associated Polya field is F(x, y) = (u(x, y), v(x, y)). A plot of the vector field can then be used to visualize the behavior of f. The complex conjugate is used rather than (u, v) itself because the Cauchy-Riemann equations for analyticity of f translate into zero divergence and zero curl for F; thus a nice interpretation of analyticity in vector field terms is available. Of course, the conjugate field preserves all important features of f. The vector field representation is ideally suited to depict the process of convergence of a sequence of complex functions. Each function f, is represented by a vector field Fn on the common domain of definition, and convergence of f, to a limiting function f can be envisioned as a stabilizing of the vector fields F11 to a limiting configuration, that of the vector field for f. Convergence at a point is thus represented by a sequence of arrows based at a common point tending toward a limiting arrow. Divergence appears as either an oscillation or a growth in magnitude without bound of a sequence of common-based arrows. The production and exploration of such sequences of vector fields is easy in an interactive computer system such as Mathernatica. We describe a program written in the Mathernatica language and implemented on the Macintosh II that allows construction of such fields for a complex power series. The program displays the partial sums of a series oo=a1a(z Z0) l in vector field form. The first input to the program is a domain, which can be a disc or an annulus with center z0 and any radii. The next input is a pair [a, b]; only vectors whose magnitudes are between a and b will be shown. Then the user gives the number of annuli into which the display will be divided, and the closed form expression for an. The program computes and displays the fields in succession; vectors are plotted as arrows with bases on circles determined by the subdivision annuli. The lengths of the vectors are scaled logarithmically to avoid messy overlaps; this does not alter the behavior of the sequence. Arrows whose lengths exceed the upper tolerance level are deleted and replaced by dots. Termination of the program leaves us with a field that may be taken as a representation of E,0a11(z Zo). The animation feature of Mathematica allows the fields to be presented in an animated sequence, showing the meaning of convergence and divergence dynamically.