1,489 research outputs found
Tur\'an type inequalities for regular Coulomb wave functions
Tur\'an, Mitrinovi\'c-Adamovi\'c and Wilker type inequalities are deduced for
regular Coulomb wave functions. The proofs are based on a Mittag-Leffler
expansion for the regular Coulomb wave function, which may be of independent
interest. Moreover, some complete monotonicity results concerning the Coulomb
zeta functions and some interlacing properties of the zeros of Coulomb wave
functions are given.Comment: 11 page
Complex-energy analysis of proton-proton fusion
An analysis of the astrophysical factor of the proton-proton weak capture
() is performed on a large energy
range covering solar-core and early Universe temperatures. The measurement of
being physically unachievable, its value relies on the theoretical
calculation of the matrix element . Surprisingly, reaches a
maximum near that has been unexplained until now. A
model-independent parametrization of valid up to about
is established on the basis of recent effective-range
functions. It provides an insight into the relationship between the maximum of
and the proton-proton resonance pole at
from analytic continuation. In addition,
this parametrization leads to an accurate evaluation of the derivatives of
, and hence of , in the limit of zero energy.Comment: 13 pages, 9 figures, 43 reference
Basic Methods for Computing Special Functions
This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are
frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website
Logarithmic perturbation theory for radial Klein-Gordon equation with screened Coulomb potentials via expansions
The explicit semiclassical treatment of logarithmic perturbation theory for
the bound-state problem within the framework of the radial Klein-Gordon
equation with attractive real-analytic screened Coulomb potentials, contained
time-component of a Lorentz four-vector and a Lorentz-scalar term, is
developed. Based upon -expansions and suitable quantization conditions a
new procedure for deriving perturbation expansions is offered. Avoiding
disadvantages of the standard approach, new handy recursion formulae with the
same simple form both for ground and excited states have been obtained. As an
example, the perturbation expansions for the energy eigenvalues for the
Hulth\'en potential containing the vector part as well as the scalar component
are considered.Comment: 14 pages, to be submitted to Journal of Physics
QQ-onia package: a numerical solution to the Schrodinger radial equation for heavy quarkonium
This paper presents the basics of the QQ-onia package, a software based upon
the Numerov method which can be used to solve the Schrodinger radial equation
using a suitable potential V(r) for the heavy quarkonium system. This package
also allows the analysis of relevant properties of those resonances such as the
wave functions at the origin, their corresponding derivatives for l \neq 0
states, typical heavy quark velocities, and mean square radii. Besides, it
includes a tool to analize the spin-dependent contributions to the heavy
quarkonia spectrum, providing the hyperfine splittings, as well as the nPJ
energy levels. Finally, a simple software implemented in QQ-onia to compute E1
transition rates is presented.Comment: 30 pages, 3 figures, a new section on spin-dependent terms in the
potential has been adde
Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method
A fourth order fixed point method to compute the zeros of solutions of second order
homogeneous linear ODEs is obtained from the approximate integration of the Riccati equation
associated with the ODE. The method requires the evaluation of the logarithmic derivative of the
function and also uses the coefficients of the ODE. An algorithm to compute with certainty all the
zeros in an interval is given which provides a fast, reliable, and accurate method of computation.
The method is illustrated by the computation of the zeros of Gauss hypergeometric functions (including
Jacobi polynomials) and confluent hypergeometric functions (Laguerre polynomials, Hermite
polynomials, and Bessel functions included) among others. The examples show that typically 4 or 5
iterations per root are enough to provide more than 100 digits of accuracy, without requiring a priori
estimations of the roots
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