Related papers: On the Bessel solution of Kepler's equation
Since more than three centuries Kepler's equation continues to represents an important benchmark for testing new computational techniques. In the present paper, the classical Kapteyn series solution of Kepler's equation originally conceived…
This work is devoted to the obtaining of a new numerical scheme based in quadrature formulas for the Lebesgue-Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically…
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
The solution of the geodesic problem for an oblate ellipsoid is developed in terms of series. Tables are provided to simplify the computation. [This is an English translation of F. W. Bessel, Astronomische Nachrichten 4(86), 241-254 (1825).…
Euler's solution in 1734 of the Basel problem, which asks for a closed form expression for the sum of the reciprocals of all perfect squares, is one of the most celebrated results of mathematical analysis. In the modern era, numerous proofs…
The existence of elliptic periodic solutions of a perturbed Kepler problem is proved. The equations are in the plane and the perturbation depends periodically on time. The proof is based on a local description of the symplectic group in two…
The manuscripts provides a novel starting guess for the solution of Kepler's equation for unknown eccentric anomaly E given the eccentricity e and mean anomaly M of an elliptical orbit.
A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.
The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's…
Numerical solutions of Kepler's Equation are critical components of celestial mechanics software, and are often computation hot spots. This work uses symbolic regression and a genetic learning algorithm to find new initial guesses for…
The Basel problem consists in finding the sum of the reciprocals of the squares of the positive integers. It was finally solved in 1735 by Leonhard Euler. In this paper, we propose a simple proof based on the Weierstrass Sine product…
This article has a twofold purpose. On the one hand I would like to draw attention to some nice exercises on the Kepler laws, due to Otto Laporte from 1970. Our discussion here has a more geometric flavour than the original analytic…
The solution of the geodesic problem for an oblate ellipsoid is developed in terms of series. Tables are provided to simplify the computation. [This is a transcription of F. W. Bessel, Astronomische Nachrichten 4(86), 241-254 (1825). The…
In a previous work, we developed the idea to solve Kepler's equation with a CORDIC-like algorithm, which does not require any division, but still multiplications in each iteration. Here we overcome this major shortcoming and solve Kepler's…
Nondegenerate truncated indefinite Stieltjes moment problem in the class $\mathbf{N}_{\kappa}^{k}$ of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step-by-step algorythm,…
A fundamental relation in celestial mechanics is Kepler's equation, linking an orbit's mean anomaly to its eccentric anomaly and eccentricity. Being transcendental, the equation cannot be directly solved for eccentric anomaly by…
Ellipsephic or Kempner-like harmonic series are series of inverses of integers whose expansion in base $B$, for some $B \geq 2$, contains no occurrence of some fixed digit or some fixed block of digits. A prototypical example was proposed…
Due to its convolution nature, the Stieltjes integral equation can be diagonalized by Mellin transform. Several explicit resolvent kernels were obtained over the years, all of convolution type. The conditions on the given function under…
The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term…
This paper concerns elliptic systems of $p$-Laplace type with complex valued coefficient and source term. We extend the real valued theory of the elliptic $p$-Laplace equation to the complex valued case. We establish the existence and…