In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately
0.66274 34193 49181 58097 47420 97109 25290.
Kepler's equation M = E − ε sin E relates the mean anomaly M with the eccentric anomaly E for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for E in terms of elementary functions, but the Lagrange reversion theorem gives the solution as a power series in ε:
or in general[1][2]
Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of M other than a multiple of π if the eccentricity exceeds a certain value that does not depend on M. The Laplace limit is this value. It is the radius of convergence of the power series.
It is the unique real solution of the transcendental equation[3]
No closed-form expression or infinite series is known for the Laplace limit.
^Finch (2003), §4.8
^Moulton (1914), §99
^Weisstein, Eric W. "Laplace Limit". MathWorld. Retrieved 18 June 2024.
In mathematics, the Laplacelimit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the...
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable...
theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called...
same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is...
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean...
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete...
process, not an isothermal process). This error was later rectified by Laplace. During the 17th century there were several attempts to measure the speed...
sequence, related to growth by recursion. Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers. The golden ratio has the...
succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. The formula is still used...
Weisstein, Eric W. "Apéry's Constant". MathWorld. Weisstein, Eric W. "LaplaceLimit". MathWorld. Weisstein, Eric W. "Soldner's Constant". MathWorld. Weisstein...
{\varphi (s)}{s}}\,ds} . The limit appearing in the integral is also taken in the sense of (tempered) distributions. The Laplace transform of the Heaviside...
49, 55, 59, 220, and 310. A speed limit of 60 mph is posted on I-10 in Lake Charles, Baton Rouge, and from LaPlace to New Orleans, I-12 in Baton Rouge...
the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign...
dividing the Laplace transform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by the Laplace transform of...
impulse responses. The transfer function is the Laplace transform of the impulse response. The Laplace transform of a system's output may be determined...
is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally...
Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes...