The Beauty of Blaschke Products

Ulrich Daepp (), Pamela Gorkin (), Gunter Semmler () and Elias Wegert ()
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Ulrich Daepp: Bucknell University
Pamela Gorkin: Bucknell University
Gunter Semmler: Technische Universität Bergakademie Freiberg
Elias Wegert: Technische Universität Bergakademie Freiberg

Chapter 3 in Handbook of the Mathematics of the Arts and Sciences, 2021, pp 45-78 from Springer

Abstract: Abstract This chapter is dedicated to showing how visual tools, using geometry and color, can be used to enhance the understanding of statements about complex functions. In particular, the focus will be on the class of finite Blaschke products, and the relevant geometries are Euclidean as well as hyperbolic. Some of the geometric tools that will be used include symmetry, tilings, and curves generated by lines constructed using particular properties of Blaschke products. The focus then turns to the possibility of visualizing when a Blaschke product is the composition of two (nontrivial) Blaschke products. Color appears in the phase portraits that are constructed, and the main results are then validated with these visual tools. This chapter begins with an introduction to finite Blaschke products, the Poincaré disk model, and phase portraits of complex functions.

Keywords: Blaschke product; Phase portrait; Ellipse; Poincaré disk model; Hyperbolic geometry; Symmetry; Composition of functions; Complex function (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-57072-3_88

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DOI: 10.1007/978-3-319-57072-3_88

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