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.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
In contrast to sets, multisets may contain elements repeatedly.
References
Amidror I (2007) The theory of the Moiré Phenomenon: Volume II: Aperiodic Layers. Springer, Dordrecht
Amidror I (2009) The theory of the Moiré Phenomenon: Volume I: Periodic Layers. Springer, London
Anderson JW (2005) Hyperbolic geometry. Springer undergraduate mathematics series, 2nd edn. Springer, London
Arnold DN, Rogness J (2008) Möbius transformations revealed. Notices Am Math Soc 55(10):1226–1231
Baldus R (1944) Nichteuklidische Geometrie: hyperbolische Geometrie der Ebene, 2nd edn. Sammlung Göschen, De Gruyter
Chalendar I, Gorkin P, Partington JR, Ross WT (2018) Clark measures and a theorem of Ritt. Math Scand 122(2):277–298. https://doi.org/10.7146/math.scand.a-104444
Cowen CC (2012) Finite Blaschke products as compositions of other finite Blaschke products. arXiv:1207.4010
Daepp U, Gorkin P, Mortini R (2002) Ellipses and finite Blaschke products. Am Math Mon 109(9):785–795. https://doi.org/10.2307/3072367
Daepp U, Gorkin P, Shaffer A, Sokolowsky B, Voss K (2015) Decomposing finite Blaschke products. J Math Anal Appl 426(2):1201–1216. https://doi.org/10.1016/j.jmaa.2015.01.039
Daepp U, Gorkin P, Shaffer A, Voss K (2018) Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other. The Carus mathematical monographs, No. 34. The Mathematical Association of America/American Mathematical Association, Providence
Euclid (1956) The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix. Dover Publications, Inc., New York, translated with introduction and commentary by Thomas L. Heath, 2nd edn
Farris FA (1998) Reviews: Visual Complex Analysis. Am Math Mont 105(6):570–576. https://doi.org/10.2307/2589427
Farris FA (2015) Creating symmetry: The Artful Mathematics of Wallpaper Patterns. Princeton University Press, Princeton. https://doi.org/10.1515/9781400865673
Fujimura M (2013) Inscribed ellipses and Blaschke products. Comput Methods Funct Theory 13(4):557–573. https://doi.org/10.1007/s40315-013-0037-8
Garcia SR, Mashreghi J, Ross WT (2018) Finite Blaschke products and their connections. Springer, Cham. https://doi.org/10.1007/978-3-319-78247-8
Garnett JB (1981) Bounded analytic functions, Pure and Applied Mathematics, vol 96. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York/London
Gorkin P, Wagner N (2017) Ellipses and compositions of finite Blaschke products. J Math Anal Appl 445(2):1354–1366. https://doi.org/10.1016/j.jmaa.2016.01.067
Gorkin P, Laroco L, Mortini R, Rupp R (1994) Composition of inner functions. Results Math 25(3-4):252–269. https://doi.org/10.1007/BF03323410
Greenberg MJ (2007) Euclidean and non-euclidean geometries: development and history, 4th edn. W. H. Freeman, New York
Heins M (1962) On a class of conformal metrics. Nagoya Math J 21:1–60
Heins M (1986) Some characterizations of finite Blaschke products of positive degree. J Analyse Math 46:162–166. https://doi.org/10.1007/BF02796581
Huttenlauch E (2013) Art as an explanatory model of universal complexity: on Carsten Nicolai’s unidisplay, in: Carsten Nicolai, unidisplay, pp 9–13 (80), Gestalten, Berlin
Jahnke E, Emde F (1909) Funktionentafeln mit Formeln und Kurven. Mathematisch-physikalische Schriften für Ingenieure und Studierende, B.G. Teubner, Leipzig und Berlin
Kraus D, Roth O (2008) Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville’s theorem. J Lond Math Soc (2) 77(1):183–202. https://doi.org/10.1112/jlms/jdm095
Kraus D, Roth O (2013) Critical points, the Gauss curvature equation and Blaschke products. In: Blaschke products and their applications. Fields institute communications, vol 65, Springer, New York, pp 133–157. https://doi.org/10.1007/978-1-4614-5341-3_7
Milnor J (1982) Hyperbolic geometry: the first 150 years. Bull Am Math Soc 6(1):9–24
Ng TW, Tsang CY (2013) Polynomials versus finite Blaschke products. In: Blaschke products and their applications. Fields institute communications, vol 65. Springer, New York, pp 249–273. https://doi.org/10.1007/978-1-4614-5341-3_14
Nicolai C (2010a) Grid Index. Die Gestalten Verlag, Berlin
Nicolai C (2010b) Moiré Index. Die Gestalten Verlag, Berlin
Ritt JF (1922) Prime and composite polynomials. Trans Am Math Soc 23(1):51–66. https://doi.org/10.2307/1988911
Semmler G, Wegert E (2019) Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems. Anal Math Phys 9(1):221–249. https://doi.org/10.1007/s13324-017-0193-5
Singer DA (2006) The location of critical points of finite Blaschke products. Conform Geom Dyn 10:117–124. https://doi.org/10.1090/S1088-4173-06-00145-7
Stäckel P (1933) Gauß als Geometer. In: Carl Friedrich Gauß Werke Band X.2 Abhandlungen über Gauss’ wissenschaftliche Tätigkeit auf den Gebieten der reinen Mathematik und Mechanik. Springer Berlin
Walsh JL (1950) The location of critical points of analytic and harmonic functions. Colloquium Publications, vol 34. American Mathematical Society
Walsh JL (1952) Note on the location of zeros of extremal polynomials in the non-euclidean plane. Acad Serbe Sci, Publ Inst Math 4:157–160
Wegert E (2011) Phase diagrams of meromorphic functions. Comput Methods Funct Theory 10(2):639–661
Wegert E (2012) Visual complex functions: An Introduction with Phase Portraits. Birkhäuser/Springer Basel AG, Basel. https://doi.org/10.1007/978-3-0348-0180-5
Wegert E, Semmler G (2011) Phase plots of complex functions: a journey in illustration. Notices Am Math Soc 58(6):768–780
Acknowledgements
Since August 2018, Pamela Gorkin has been serving as a Program Director in the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and as a component of this position, she received support from NSF for research, which included work on this paper. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this entry
Cite this entry
Daepp, U., Gorkin, P., Semmler, G., Wegert, E. (2020). The Beauty of Blaschke Products. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_88-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-70658-0_88-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70658-0
Online ISBN: 978-3-319-70658-0
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering