36,382 research outputs found
Analytic Evaluation of Four-Particle Integrals with Complex Parameters
The method for analytic evaluation of four-particle integrals, proposed by
Fromm and Hill, is generalized to include complex exponential parameters. An
original procedure of numerical branch tracking for multiple valued functions
is developed. It allows high precision variational solution of the Coulomb
four-body problem in a basis of exponential-trigonometric functions of
interparticle separations. Numerical results demonstrate high efficiency and
versatility of the new method.Comment: 13 pages, 4 figure
Asymptotic expansion of n-dimensional Faxén-type integrals
The asymptotic expansion of n-dimensional extensions of Faxén’s integral In(z) are derived for large complex values of the variable z. The theory relies on the asymptotics of the generalised hypergeometric, orWright, function. The coefficients in the exponential expansion are obtained by means of an algorithm applicable for arbitrary n. Numerical examples are given to illustrate the accuracy of the expansions
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
Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor
We present a mathematica package that performs the symbolic calculation of
integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_{\nu} (x) j_{\mu} (x) dx
where and denote spherical Bessel functions of
integer orders, with and . With the real parameter
and the integer , convergence of the integral requires that . The package provides analytical result for the integral in its most
simplified form. The novel symbolic method employed enables the calculation of
a large number of integrals of the above form in a fraction of the time
required for conventional numerical and Mathematica based brute-force methods.
We test the accuracy of such analytical expressions by comparing the results
with their numerical counterparts.Comment: 17 pages; updated references for the introductio
On the Number of Isolated Zeros of Pseudo-Abelian Integrals: Degeneracies of the Cuspidal Type
We consider a multivalued function of the form
, which is a Darboux
first integral of polynomial one-form
. We assume, for
, that the polycyle has only cuspidal singularity
which we assume at the origin and other singularities are saddles.
We consider families of Darboux first integrals unfolding
(and its cuspidal point) and pseudo-Abelian integrals associated to these
unfolding. Under some conditions we show the existence of uniform local bound
for the number of zeros of these pseudo-Abelian integrals
Theory and computation of electromagnetic transition matrix elements in the continuous spectrum of atoms
The present study examines the mathematical properties of the free-free (
f-f) matrix elements of the full electric field operator, of the multipolar
Hamiltonian. Special methods are developed and applied for their computation,
for the general case where the scattering wavefunctions are calculated
numerically in the potential of the term-dependent (N-1) electron core, and are
energy-normalized. It is found that, on the energy axis, the f-f matrix
elements of the full operator have singularities of first order in the case of
equal photoelectron energies. The numerical applications are for f-f
transitions in Hydrogen and Neon, obeying electric dipole and quadrupole
selection rules. In the limit of zero photon wave-number, the full operator
reduces to the length form of the electric dipole approximation (EDA). It is
found that the results for the EDA agree with those of the full operator, with
the exception of a photon wave-number region about the singularity.Comment: 39 pages, 11 figure
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
Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments
We describe a variety of methods to compute the functions ,
and their derivatives for real and positive . These
functions are numerically satisfactory independent solutions of the
differential equation . In an accompanying paper
(Algorithm xxx: Modified Bessel functions of imaginary order and positive
argument) we describe the implementation of these methods in Fortran 77 codes.Comment: 14 pages, 1 figure. To appear in ACM T. Math. Sof
- …