On regularization of second kind integrals
Loading...
Date
2018
Authors
Bernatska, Julia
Leykin, Dmitry
Journal Title
Journal ISSN
Volume Title
Publisher
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.
Description
Keywords
second kind integral, regularization constant, Abelian function relation, Jacobi inversion problem, addition formula, article
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