Infinite Product Representation for the Szegö Kernel for an Annulus
The Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2022/3763450 |
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| author | Nuraddeen S. Gafai Ali H. M. Murid Nur H. A. A. Wahid |
| author_facet | Nuraddeen S. Gafai Ali H. M. Murid Nur H. A. A. Wahid |
| author_sort | Nuraddeen S. Gafai |
| collection | DOAJ |
| description | The Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic bilateral series (also known as q-bilateral series). This leads to an infinite product representation through the application of Ramanujan’s sum. The infinite product clearly exhibits the unique zero of the Szegö kernel for an annulus. Its connection with the basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szegö kernel for general annulus and weighted Szegö kernel. Numerical comparisons on computing the Szegö kernel for an annulus based on the Kerzman-Stein integral equation, the bilateral series, and the infinite product are also presented. |
| format | Article |
| id | doaj-art-0b84ae2ed91846258a0a41383011a0d2 |
| institution | Kabale University |
| issn | 2314-8888 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-0b84ae2ed91846258a0a41383011a0d22025-02-03T01:06:32ZengWileyJournal of Function Spaces2314-88882022-01-01202210.1155/2022/3763450Infinite Product Representation for the Szegö Kernel for an AnnulusNuraddeen S. Gafai0Ali H. M. Murid1Nur H. A. A. Wahid2Department of Mathematics and StatisticsDepartment of Mathematical SciencesFaculty of Computer and Mathematical SciencesThe Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic bilateral series (also known as q-bilateral series). This leads to an infinite product representation through the application of Ramanujan’s sum. The infinite product clearly exhibits the unique zero of the Szegö kernel for an annulus. Its connection with the basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szegö kernel for general annulus and weighted Szegö kernel. Numerical comparisons on computing the Szegö kernel for an annulus based on the Kerzman-Stein integral equation, the bilateral series, and the infinite product are also presented.http://dx.doi.org/10.1155/2022/3763450 |
| spellingShingle | Nuraddeen S. Gafai Ali H. M. Murid Nur H. A. A. Wahid Infinite Product Representation for the Szegö Kernel for an Annulus Journal of Function Spaces |
| title | Infinite Product Representation for the Szegö Kernel for an Annulus |
| title_full | Infinite Product Representation for the Szegö Kernel for an Annulus |
| title_fullStr | Infinite Product Representation for the Szegö Kernel for an Annulus |
| title_full_unstemmed | Infinite Product Representation for the Szegö Kernel for an Annulus |
| title_short | Infinite Product Representation for the Szegö Kernel for an Annulus |
| title_sort | infinite product representation for the szego kernel for an annulus |
| url | http://dx.doi.org/10.1155/2022/3763450 |
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