25 research outputs found
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Resummation of Nonalternating Divergent Perturbative Expansions
A method for the resummation of nonalternating divergent perturbation series
is described. The procedure constitutes a generalization of the Borel-Pad\'{e}
method. Of crucial importance is a special integration contour in the complex
plane. Nonperturbative imaginary contributions can be inferred from the purely
real perturbative coefficients. A connection is drawn from the quantum field
theoretic problem of resummation to divergent perturbative expansions in other
areas of physics.Comment: 5 pages, LaTeX, 2 tables, 1 figure; discussion of the Carleman
criterion added; version to appear in Phys. Rev.
Dressed States Approach to Quantum Systems
Using the non-perturbative method of {\it dressed} states previously
introduced in JPhysA, we study effects of the environment on a quantum
mechanical system, in the case the environment is modeled by an ensemble of non
interacting harmonic oscillators. This method allows to separate the whole
system into the {\it dressed} mechanical system and the {\it dressed}
environment, in terms of which an exact, non-perturbative approach is possible.
When applied to the Brownian motion, we give explicit non-perturbative formulas
for the classical path of the particle in the weak and strong coupling regimes.
When applied to study atomic behaviours in cavities, the method accounts very
precisely for experimentally observed inhibition of atomic decay in small
cavities PhysLA, physics0111042
Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field
We consider the resummation of the perturbation series describing the energy
displacement of a hydrogenic bound state in an electric field (known as the
Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal
power series in the electric field strength. The perturbation series exhibits a
rich singularity structure in the Borel plane. Resummation methods are
presented which appear to lead to consistent results even in problematic cases
where isolated singularities or branch cuts are present on the positive and
negative real axis in the Borel plane. Two resummation prescriptions are
compared: (i) a variant of the Borel-Pade resummation method, with an
additional improvement due to utilization of the leading renormalon poles (for
a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. vol. 317,
p. 1 (1999)]), and (ii) a contour-improved combination of the Borel method with
an analytic continuation by conformal mapping, and Pade approximations in the
conformal variable. The singularity structure in the case of the LoSurdo-Stark
effect in the complex Borel plane is shown to be similar to (divergent)
perturbative expansions in quantum chromodynamics.Comment: 14 pages, RevTeX, 3 tables, 1 figure; numerical accuracy of results
enhanced; one section and one appendix added and some minor changes and
additions; to appear in phys. rev.
QED Effective Action Revisited
The derivation of a convergent series representation for the quantum
electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in
[Can. J. Phys. vol. 71, p. 389 (1993)] is reexamined. We present more details
of our original derivation. Moreover, we discuss the relation of the
electric-magnetic duality to the integral representation for the effective
action, and we consider the application of nonlinear convergence acceleration
techniques which permit the efficient and reliable numerical evaluation of the
quantum correction to the Maxwell Lagrangian.Comment: 20 pages, LaTeX, 1 table; minor additions and adjustments; to appear
in Can. J. Phy
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