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Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions Let $$f=B_1/B_2$$ f = B 1 / B 2 be a ratio of finite Blaschke products having no critical points on $$\partial \mathbb {D}$$ ∂ D . Then $$f$$ f has finitely many critical level curves (level curves containing critical points of $$f$$ f ) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $$f$$ f , one needs only understand the configuration of the finitely many critical level curves of $$f$$ f . In this paper, we show that in fact such a function $$f$$ f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $$f_1$$ f 1 and $$f_2$$ f 2 have the same configuration of critical level curves, then there is a conformal map $$\phi $$ ϕ such that $$f_1\equiv f_2\circ \phi $$ f 1 ≡ f 2 ∘ ϕ . We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-015-0111-5
Publisher site
See Article on Publisher Site

Abstract

Let $$f=B_1/B_2$$ f = B 1 / B 2 be a ratio of finite Blaschke products having no critical points on $$\partial \mathbb {D}$$ ∂ D . Then $$f$$ f has finitely many critical level curves (level curves containing critical points of $$f$$ f ) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $$f$$ f , one needs only understand the configuration of the finitely many critical level curves of $$f$$ f . In this paper, we show that in fact such a function $$f$$ f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $$f_1$$ f 1 and $$f_2$$ f 2 have the same configuration of critical level curves, then there is a conformal map $$\phi $$ ϕ such that $$f_1\equiv f_2\circ \phi $$ f 1 ≡ f 2 ∘ ϕ . We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 31, 2015

References