Complex Functions and Images
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Complex Functions and Images Elias Wegert
Received: 3 December 2012 / Accepted: 15 December 2012 / Published online: 13 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract The cover of this volume shows the phase portrait of a rational function. In this note, we explain how its poles and zeros are chosen in order to create the four letters C, M, F, T. Moreover, we prove that phase portraits of rational functions can “visually approximate” any image composed of saturated colors. Keywords Complex functions · Visualization · Phase portrait · Domain coloring · Approximation Mathematics Subject Classification (2000)
Primary 41A20; Secondary 30E10
1 Introduction In recent years, the visualization of complex (analytic) functions has attracted considerable interest (see [1,3–6,8,10,12] and the references therein). Besides the classical grid mappings, various color representations depicting the color-coded values of a function directly on its domain became quite popular. While standard domain coloring ([3,5]) encodes the complete values of a function using a two-dimensional color scheme, so-called phase portraits or phase plots ([10], [12]) depict only the phase (argument) of the function. The three windows of the Fig. 1 show a standard domain-coloring (left), a proper phase portrait (middle), and an enhanced phase portrait of the complex sine function in the square |Re z| < 5, |Im z| < 5. All these three representations have advantages Communicated by Stephan Ruscheweyh. Supported by the Deutsche Forschungsgemeinschaft, grant We1704/8-2. E. Wegert (B) Institute of Applied Analysis, TU Bergakademie Freiberg, Akademiestr. 6, 09596 Freiberg, Germany e-mail: [email protected]
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Fig. 1 Three color representations of the complex sine function
Fig. 2 The hsv color-wheel, the phase in a square, and a phase portrait
and drawbacks—which version one prefers is not only a matter of taste but depends on the purpose for which such an image is created. 2 Phase Portraits and Their Modifications In order to avoid the multi-valuedness of the argument of a complex number z, we prefer to work with its phase, defined for z ∈ C\{0} by ψ(z) := z/|z|. Since the phase attains values on the complex unit circle T, it can be conveniently encoded using colors from the standard hsv color-wheel shown in Fig. 2 (left), to which we refer as saturated colors. Defining ψ(0) := 0 (represented by black) and ψ(∞) := ∞ (represented by white) extends phase to all points of the Riemann sphere C. The picture in the middle of Fig. 2 shows the color-coded phase of points in a square centered at the origin. As a mathematical entity, the phase portrait of a function f : D → C is just the mapping ψ ◦ f . Depicting the color-coded values of ψ ◦ f on D (or on a subset thereof) yields an image which we also call a phase portrait of f . The image on the right of Fig. 2 is the phase portrait of the rational function f (z) = (z − 1)/(z 2 + z + 1) in the square |Re z| < 2, |Im z| < 2. Note that zeros and poles of f can be distin
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