On regularization of second kind integrals

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dc.contributor.author Bernatska, Julia
dc.contributor.author Leykin, Dmitry
dc.date.accessioned 2019-06-06T09:37:29Z
dc.date.available 2019-06-06T09:37:29Z
dc.date.issued 2018
dc.identifier.citation Bernatska J. On regularization of second kind integrals [electronic resource] / Bernatska J., Leykin D. // Symmetry, integrability and geometry, methods and applications : SIGMA. - 2018. - Vol. 14. - Article number 074. - DOI: 10.3842/SIGMA.2018.074 en_US
dc.identifier.issn 1815-0659
dc.identifier.uri http://ekmair.ukma.edu.ua/handle/123456789/15854
dc.identifier.uri doi.org/10.3842/SIGMA.2018.074
dc.description.abstract 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 basepoint for Abel’s map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the 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 Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees 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 of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used. en_US
dc.language.iso en en_US
dc.subject second kind integral en_US
dc.subject regularization constant en_US
dc.subject Abelian function relation en_US
dc.subject Jacobi inversion problem en_US
dc.subject addition formula en_US
dc.subject article en_US
dc.title On regularization of second kind integrals en_US
dc.type Article en_US
dc.status published earlier en_US
dc.relation.source Symmetry, integrability and geometry, methods and applications : SIGMA. - 2018. - Vol. 14 en_US

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