2,245 research outputs found
Numerical computation of real or complex elliptic integrals
Algorithms for numerical computation of symmetric elliptic integrals of all
three kinds are improved in several ways and extended to complex values of the
variables (with some restrictions in the case of the integral of the third
kind). Numerical check values, consistency checks, and relations to Legendre's
integrals and Bulirsch's integrals are included
Theory for planetary exospheres: I. Radiation pressure effect on dynamical trajectories
The planetary exospheres are poorly known in their outer parts, since the
neutral densities are low compared with the instruments detection capabilities.
The exospheric models are thus often the main source of information at such
high altitudes. We present a new way to take into account analytically the
additional effect of the radiation pressure on planetary exospheres. In a
series of papers, we present with an Hamiltonian approach the effect of the
radiation pressure on dynamical trajectories, density profiles and escaping
thermal flux. Our work is a generalization of the study by Bishop and
Chamberlain (1989). In this first paper, we present the complete exact
solutions of particles trajectories, which are not conics, under the influence
of the solar radiation pressure. This problem was recently partly solved by
Lantoine and Russell (2011) and completely by Biscani and Izzo (2014). We give
here the full set of solutions, including solutions not previously derived, as
well as simpler formulations for previously known cases and comparisons with
recent works. The solutions given may also be applied to the classical Stark
problem (Stark,1914): we thus provide here for the first time the complete set
of solutions for this well-known effect in term of Jacobi elliptic functions
Analytical solutions of bound timelike geodesic orbits in Kerr spacetime
We derive the analytical solutions of the bound timelike geodesic orbits in
Kerr spacetime. The analytical solutions are expressed in terms of the elliptic
integrals using Mino time as the independent variable. Mino time
decouples the radial and polar motion of a particle and hence leads to forms
more useful to estimate three fundamental frequencies, radial, polar and
azimuthal motion, for the bound timelike geodesics in Kerr spacetime. This
paper gives the first derivation of the analytical expressions of the
fundamental frequencies. This paper also gives the first derivation of the
analytical expressions of all coordinates for the bound timelike geodesics
using Mino time. These analytical expressions should be useful not only to
investigate physical properties of Kerr geodesics but more importantly to
applications related to the estimation of gravitational waves from the extreme
mass ratio inspirals.Comment: A typo in the first expression in equation 21 was fixe
Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology
Accurate and efficient methods to evaluate cosmological distances are an
important tool in modern precision cosmology. In a flat CDM cosmology,
the luminosity distance can be expressed in terms of elliptic integrals. We
derive an alternative and simple expression for the luminosity distance in a
flat CDM based on hypergeometric functions. Using a timing experiment
we compare the computation time for the numerical evaluation of the various
exact formulae, as well as for two approximate fitting formulae available in
the literature. We find that our novel expression is the most efficient exact
expression in the redshift range . Ideally, it can be combined with
the expression based on Carlson's elliptic integrals in the range
for high precision cosmology distance calculations over the entire redshift
range. On the other hand, for practical work where relative errors of about
0.1% are acceptable, the analytical approximation proposed by Adachi & Kasai
(2012) is a suitable alternative.Comment: 4 pages, 1 figure, accepted for publication in MNRA
Transverse Mercator with an accuracy of a few nanometers
Implementations of two algorithms for the transverse Mercator projection are
described; these achieve accuracies close to machine precision. One is based on
the exact equations of Thompson and Lee and the other uses an extension of
Krueger's series for the projection to higher order. The exact method provides
an accuracy of 9 nm over the entire ellipsoid, while the errors in the series
method are less than 5 nm within 3900 km of the central meridian. In each case,
the meridian convergence and scale are also computed with similar accuracy. The
speed of the series method is competitive with other less accurate algorithms
and the exact method is about 5 times slower.Comment: LaTeX, 10 pages, 3 figures. Includes some revisions. Supplementary
material is available at http://geographiclib.sourceforge.net/tm.htm
Series expansions for the third incomplete elliptic integral via partial fraction decompositions
We find convergent double series expansions for Legendre's third incomplete
elliptic integral valid in overlapping subdomains of the unit square. Truncated
expansions provide asymptotic approximations in the neighbourhood of the
logarithmic singularity if one of the variables approaches this point
faster than the other. Each approximation is accompanied by an error bound. For
a curve with an arbitrary slope at our expansions can be rearranged
into asymptotic expansions depending on a point on the curve. For reader's
convenience we give some numeric examples and explicit expressions for
low-order approximations.Comment: The paper has been substantially updated (hopefully improved) and
divided in two parts. This part is about third incomplete elliptic integral.
10 page
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Polar harmonic Maass forms and their applications
In this survey, we present recent results of the authors about
non-meromorphic modular objects known as polar harmonic Maass forms. These
include the computation of Fourier coefficients of meromorphic modular forms
and relations between inner products of meromorphic modular forms and higher
Green's functions evaluated at CM-points
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