

Seminar on Complex Analysis (Gonchar Seminar)
December 16, 2013 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)






Computation of Abelian integrals without quadratures
A. B. Bogatyrev^{} ^{} Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow

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Abstract:
Formulas that go back to Riemann, represent abelian integrals in terms of theta functions. However, these formulas explicitly contain the AbelJacobi mapping, that is Abelian integrals again. One can circumvent the stumbling block by explicit localization of the image of the curve inside its Jacobian under the AJ map. We now have a parametric representation of abelian integrals: an explicit expression for the integral on the one hand, and an explicit expression for some projection of the curve onto the sphere  on the other. This representation of the Abelian integral can be used either for its computation (with the machine precision), or  for its inversion.

