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Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions

Abstract

We exploit the equality of Bergman analytic content and torsional rigidity of a simply connected domain to develop a new method for calculating these quantities. This method is particularly suitable for the case when the region in question is a polygon. A large number of examples are computed in explicit detail to demonstrate the utility of our ideas.

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Notes

  1. We note that (5) is valid for any domain in \(\varOmega \subseteq \mathbb {C}\) for which the Bergman polynomials form a basis for the Bergman space of \(\varOmega \).

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Acknowledgements

The authors would like to thank D. Khavinson for useful feedback on a first draft of this manuscript.

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Correspondence to Brian Simanek.

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Communicated by Dmitry Khavinson.

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Fleeman, M., Simanek, B. Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions. Comput. Methods Funct. Theory 19, 37–63 (2019). https://doi.org/10.1007/s40315-018-0252-4

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  • DOI: https://doi.org/10.1007/s40315-018-0252-4

Keywords

  • Torsional rigidity
  • Bergman analytic content
  • Bergman polynomials

Mathematics Subject Classification

  • 30E10
  • 30H20