102,541 research outputs found
Computing the Lambert W function in arbitrary-precision complex interval arithmetic
We describe an algorithm to evaluate all the complex branches of the Lambert
W function with rigorous error bounds in interval arithmetic, which has been
implemented in the Arb library. The classic 1996 paper on the Lambert W
function by Corless et al. provides a thorough but partly heuristic numerical
analysis which needs to be complemented with some explicit inequalities and
practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure
On evaluation of the Heun functions
In the paper we deal with the Heun functions --- solutions of the Heun
equation, which is the most general Fuchsian equation of second order with four
regular singular points. Despite the increasing interest to the equation and
numerous applications of the functions in a wide variety of physical problems,
it is only Maple amidst known software packages which is able to evaluate the
Heun functions numerically. But the Maple routine is known to be imperfect:
even at regular points it may return infinities or end up with no result.
Improving the situation is difficult because the code is not publicly
available. The purpose of the work is to suggest and develop alternative
algorithms for numerical evaluation of the Heun functions. A procedure based on
power series expansions and analytic continuation is suggested which allows us
to avoid numerical integration of the differential equation and to achieve
reasonable efficiency and accuracy. Results of numerical tests are given
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
Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
We study the analytic continuation of Feynman integrals from the kite family,
expressed in terms of elliptic generalisations of (multiple) polylogarithms.
Expressed in this way, the Feynman integrals are functions of two periods of an
elliptic curve. We show that all what is required is just the analytic
continuation of these two periods. We present an explicit formula for the two
periods for all values of . Furthermore, the nome of the
elliptic curve satisfies over the complete range in the inequality , where is attained only at the singular points
. This ensures the convergence of the -series
expansion of the -functions and provides a fast and efficient
evaluation of these Feynman integrals.Comment: 30 pages, version to be publishe
RacoonWW1.3: A Monte Carlo program for four-fermion production at e^+ e^- colliders
We present the Monte Carlo generator RacoonWW that computes cross sections to
all processes e^+ e^- -> 4f and e^+ e^- -> 4f + gamma and calculates the
complete O(alpha) electroweak radiative corrections to e^+ e^- -> W W -> 4f in
the electroweak Standard Model in double-pole approximation. The calculation of
the tree-level processes e^+ e^- -> 4f and e^+ e^- -> 4f + gamma is based on
the full matrix elements for massless (polarized) fermions. When calculating
radiative corrections to e^+ e^- -> W W -> 4f the complete virtual
doubly-resonant electroweak corrections are included, i.e. the factorizable and
non-factorizable virtual corrections in double-pole approximation, and the real
corrections are based on the full matrix elements for e^+ e^- -> 4f + gamma.
The matching of soft and collinear singularities between virtual and real
corrections is done alternatively in two different ways, namely by using a
subtraction method or by applying phase-space slicing. Higher-order
initial-state photon radiation and naive QCD corrections are taken into
account. RacoonWW also provides anomalous triple gauge-boson couplings for all
processes e^+ e^- -> 4f and anomalous quartic gauge-boson couplings for all
processes e^+ e^- -> 4f + gamma.Comment: 62 pages, LaTeX, elsart styl
Rigorous Born Approximation and beyond for the Spin-Boson Model
Within the lowest-order Born approximation, we present an exact calculation
of the time dynamics of the spin-boson model in the ohmic regime. We observe
non-Markovian effects at zero temperature that scale with the system-bath
coupling strength and cause qualitative changes in the evolution of coherence
at intermediate times of order of the oscillation period. These changes could
significantly affect the performance of these systems as qubits. In the biased
case, we find a prompt loss of coherence at these intermediate times, whose
decay rate is set by , where is the coupling strength
to the environment. We also explore the calculation of the next order Born
approximation: we show that, at the expense of very large computational
complexity, interesting physical quantities can be rigorously computed at
fourth order using computer algebra, presented completely in an accompanying
Mathematica file. We compute the corrections to the long time
behavior of the system density matrix; the result is identical to the reduced
density matrix of the equilibrium state to the same order in . All
these calculations indicate precision experimental tests that could confirm or
refute the validity of the spin-boson model in a variety of systems.Comment: Greatly extended version of short paper cond-mat/0304118.
Accompanying Mathematica notebook fop5.nb, available in Source, is an
essential part of this work; it gives full details of the fourth-order Born
calculation summarized in the text. fop5.nb is prepared in arXiv style
(available from Wolfram Research
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