On regularization of second kind integrals

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dc.contributor.authorBernatska, Julia
dc.contributor.authorLeykin, Dmitry
dc.date.accessioned2019-06-06T09:37:29Z
dc.date.available2019-06-06T09:37:29Z
dc.date.issued2018
dc.description.abstractWe 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.identifier.citationBernatska 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.2018.074en_US
dc.identifier.issn1815-0659
dc.identifier.urihttps://ekmair.ukma.edu.ua/handle/123456789/15854
dc.identifier.uridoi.org/10.3842/SIGMA.2018.074
dc.language.isoenen_US
dc.relation.sourceSymmetry, integrability and geometry, methods and applications : SIGMA. - 2018. - Vol. 14en_US
dc.statuspublished earlieren_US
dc.subjectsecond kind integralen_US
dc.subjectregularization constanten_US
dc.subjectAbelian function relationen_US
dc.subjectJacobi inversion problemen_US
dc.subjectaddition formulaen_US
dc.subjectarticleen_US
dc.titleOn regularization of second kind integralsen_US
dc.typeArticleen_US
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