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Analysis as a Life pp 213–235Cite as

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The String Equation for Some Rational Functions

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Abstract

For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). For non-univalent analytic functions the Poisson bracket may become ambiguous, hence the string equation need not make sense. In the present paper we show that for a certain class of (non-univalent) rational functions related to quadrature Riemann surfaces, the string equation does make sense, and holds.

Dedicated to Heinrich Begehr, on the occasion of his 80th birthday

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Acknowledgement

The author is grateful to an anonymous referee for constructive comments on the paper.

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  1. Department of Mathematics, KTH, Stockholm, Sweden

    Björn Gustafsson

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  1. Björn Gustafsson
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Correspondence to Björn Gustafsson .

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  1. Department of Economics, Belarusian State University, Minsk, Belarus

    Sergei Rogosin

  2. Mathematics Department, Yeditepe University, Ataşehir, Istanbul, Turkey

    Ahmet Okay Çelebi

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Gustafsson, B. (2019). The String Equation for Some Rational Functions. In: Rogosin, S., Çelebi, A. (eds) Analysis as a Life. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02650-9_11

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