On the integral solution of elliptic Kepler’s equation

Abstract

AbstractIn a recent paper, Philcox, Goodman and Slepian obtain an explicit solution of the elliptic Kepler’s equation (KE) as a quotient of two contour integrals along a Jordan curve $$ \mathcal {C} = \mathcal {C}(M,e)$$ C = C ( M , e ) that contains the unique real solution of KE but not includes other complex zeros of KE in its interior. The aim of this paper is to study the main issues that arise in the practical implementation of this integral solution. Thus, after a study of the complex zeros of KE, several families of Jordan contours $$ \mathcal {C} = \mathcal {C}(M,e)$$ C = C ( M , e ) that are suitable for this integral solution are proposed. Since contours with minimal length turn out to be the more accurate for numerical purposes, several families that minimize their length are constructed. Secondly, the approximation of the contour integrals by the composite trapezoidal rule is considered. Recall that this rule is employed in the fast Fourier transform and, in spite of its lower order, displays a spectral convergence as a function of the number of nodes, which implies a very fast convergence. Finally, the results of some numerical experiments are presented to show that such a combination of appropriate contours with the composite trapezoidal rule leads to a powerful numerical method to solve KE with any desired accuracy for all values of eccentricity.