On Regularization of Second Kind Integrals
We obtain expressions for second kind integrals on non-hyperelliptic (n,s)-curves. Such a curve possesses a Weierstrass point at infinity, which is a branch point where all sheets of the curve come together. The infinity serves as the base point for Abel's map, and the base point in the definit...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209776 |
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| Cite this: | On Regularization of Second Kind Integrals / J. Bernatska, D. Leykin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. |
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Bernatska, J. Leykin, D. 2025-11-26T11:40:52Z 2018 On Regularization of Second Kind Integrals / J. Bernatska, D. Leykin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32A55; 32W50; 35R01; 14H40; 32A15; 14H45 arXiv: 1709.10167 https://nasplib.isofts.kiev.ua/handle/123456789/209776 https://doi.org/10.3842/SIGMA.2018.074 We obtain expressions for second kind integrals on non-hyperelliptic (n,s)-curves. Such a curve possesses a Weierstrass point at infinity, which is a branch point where all sheets of the curve come together. The infinity serves as the base point for Abel's map, and the base point in the definition of the second kind of integrals. We define second kind differentials as having a pole at infinity; therefore, the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on the Jacobian of the curve. In this connection, we introduce the notion of a regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of infinity. This is a vector with components depending on parameters of the curve; the number of components is equal to the genus of the curve. The presence of the term guarantees the consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for (3,4), (3,5), (3,7), and (4,5)-curves. By the example of (3,4)-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme for obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used. These results were presented and discussed at seminars of the School of Mathematics at the Universities of Leeds, Loughborough, Glasgow, Edinburgh, and Heriot-Watt University. The authors are grateful to V. Enolski for stimulating discussions, A.V. Mikhailov, O. Chalykh, A.P. Veselov, C. Athorne, H.W. Braden, and J.C. Eilbeck for hospitality and substantive comments. The authors are grateful to the anonymous referees for their essential contribution to the improvement of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Regularization of Second Kind Integrals Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On Regularization of Second Kind Integrals |
| spellingShingle |
On Regularization of Second Kind Integrals Bernatska, J. Leykin, D. |
| title_short |
On Regularization of Second Kind Integrals |
| title_full |
On Regularization of Second Kind Integrals |
| title_fullStr |
On Regularization of Second Kind Integrals |
| title_full_unstemmed |
On Regularization of Second Kind Integrals |
| title_sort |
on regularization of second kind integrals |
| author |
Bernatska, J. Leykin, D. |
| author_facet |
Bernatska, J. Leykin, D. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We obtain expressions for second kind integrals on non-hyperelliptic (n,s)-curves. Such a curve possesses a Weierstrass point at infinity, which is a branch point where all sheets of the curve come together. The infinity serves as the base point for Abel's map, and the base point in the definition of the second kind of integrals. We define second kind differentials as having a pole at infinity; therefore, the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on the Jacobian of the curve. In this connection, we introduce the notion of a regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of infinity. This is a vector with components depending on parameters of the curve; the number of components is equal to the genus of the curve. The presence of the term guarantees the consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for (3,4), (3,5), (3,7), and (4,5)-curves. By the example of (3,4)-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme for obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209776 |
| citation_txt |
On Regularization of Second Kind Integrals / J. Bernatska, D. Leykin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. |
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2025-12-07T12:52:45Z |
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2025-12-07T12:52:45Z |
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1850885993228402688 |