14,784 research outputs found
A fast analysis-based discrete Hankel transform using asymptotic expansions
A fast and numerically stable algorithm is described for computing the
discrete Hankel transform of order as well as evaluating Schl\"{o}milch and
Fourier--Bessel expansions in
operations. The algorithm is based on an asymptotic expansion for Bessel
functions of large arguments, the fast Fourier transform, and the Neumann
addition formula. All the algorithmic parameters are selected from error bounds
to achieve a near-optimal computational cost for any accuracy goal. Numerical
results demonstrate the efficiency of the resulting algorithm.Comment: 22 page
An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
We describe a method for the rapid numerical evaluation of the Bessel
functions of the first and second kinds of nonnegative real orders and positive
arguments. Our algorithm makes use of the well-known observation that although
the Bessel functions themselves are expensive to represent via piecewise
polynomial expansions, the logarithms of certain solutions of Bessel's equation
are not. We exploit this observation by numerically precomputing the logarithms
of carefully chosen Bessel functions and representing them with piecewise
bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions
of orders between and 1\sep,000\sep,000\sep,000 at essentially any
positive real argument. In that regime, it is competitive with existing methods
for the rapid evaluation of Bessel functions and has several advantages over
them. First, our approach is quite general and can be readily applied to many
other special functions which satisfy second order ordinary differential
equations. Second, by calculating the logarithms of the Bessel functions rather
than the Bessel functions themselves, we avoid many issues which arise from
numerical overflow and underflow. Third, in the oscillatory regime, our
algorithm calculates the values of a nonoscillatory phase function for Bessel's
differential equation and its derivative. These quantities are useful for
computing the zeros of Bessel functions, as well as for rapidly applying the
Fourier-Bessel transform. The results of extensive numerical experiments
demonstrating the efficacy of our algorithm are presented. A Fortran package
which includes our code for evaluating the Bessel functions as well as our code
for all of the numerical experiments described here is publically available
Geometric factors in the Bohr--Rosenfeld analysis of the measurability of the electromagnetic field
The Geometric factors in the field commutators and spring constants of the
measurement devices in the famous analysis of the measurability of the
electromagnetic field by Bohr and Rosenfeld are calculated using a
Fourier--Bessel method for the evaluation of folding integrals, which enables
one to obtain the general geometric factors as a Fourier--Bessel series. When
the space region over which the factors are defined are spherical, the
Fourier--Bessel series terms are given by elementary functions, and using the
standard Fourier-integral method of calculating folding integrals, the
geometric factors can be evaluated in terms of manageable closed-form
expressions.Comment: 21 pages, REVTe
On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
We review recently developed new powerful techniques to compute a class of
Feynman diagrams at any loop order, known as sunrise-type diagrams. These
sunrise-type topologies have many important applications in many different
fields of physics and we believe it to be timely to discuss their evaluation
from a unified point of view. The method is based on the analysis of the
diagrams directly in configuration space which, in the case of the sunrise-type
diagrams and diagrams related to them, leads to enormous simplifications as
compared to the traditional evaluation of loops in momentum space. We present
explicit formulae for their analytical evaluation for arbitrary mass
configurations and arbitrary dimensions at any loop order. We discuss several
limiting cases of their kinematical regimes which are e.g. relevant for
applications in HQET and NRQCD.Comment: 100 pages, 16 eps-figures include
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