On the Integral Solution of Hyperbolic Kepler's  Equation

Abstract In a recent paper of Philcox, Goodman and Slepian, the solution of elliptic Kepler's equation is given as a quotient of twocontour integrals along a Jordan curve that contains in their interior the unique real solution of elliptic Kepler's equation and does not include other complex zeroes. In this paper we show that a similar explicit integral solution can be given for the hyperbolic Kepler's equation. With this purpose, we carry out a study of the complex zeros of the hyperbolic Kepler's equation in order to define suitable Jordan contours in the integrals. Even more, we show that appropriate elliptic Jordan contours can be defined for such integrals, which reduces the computing time. Moreover, following the ideas of the Fast Fourier Trnasform (FFT) algorithm, these integrals can be approximated by the composite trapezoidal rule which gives an algorithm with spectral convergence as a function of the number of nodes. The results of some numerical experiments are presented to show that this implementation is a reliable and very accurate algorithm for solving the hyperbolic Kepler's equation.