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The Szegö kernel and oblique projections: conformal mapping of non-smooth regions

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Abstract

The method of Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) for computing the Riemann mapping function of a smooth domain is extended to include the case of simply connected convex regions with corners, in particular convex polygons. The connection between the Szegö kernel and the Riemann mapping function is classical. The integral equation in Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) that determines the Szegö kernel is no longer defined in the presence of corners. We modify the equation by using new oblique projections. This approach is equivalent to employing preliminary mappings.

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Acknowledgements

In memory of Norberto Kerzman (1943–2019) with whom the author had many discussions about mathematics and life.

Funding

This work was supported by a National Science and Engineering Research Council (NSERC) Canada Discovery grant (RGPIN-2020-04663).

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  1. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Manfred R. Trummer

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  1. Manfred R. Trummer
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In this single-authored article, all material was created by the author. That includes mathematical theory and proofs, the code for computing conformal maps, as well as all figures, plots, and tables.

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Correspondence to Manfred R. Trummer.

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Trummer, M.R. The Szegö kernel and oblique projections: conformal mapping of non-smooth regions. Numer Algor 95, 929–942 (2024). https://doi.org/10.1007/s11075-023-01594-x

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