28,660 research outputs found
Asymptotic Expansions of exp-log Functions
We give an algorithm to compute asymptotic expansions of exp-log functions. This algorithm automatically computes the necessary asymptotic scale and does not suffer from problems of indefinite cancellation. In particular, an asymptotic equivalent can always be computed for a given exp-log function
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
Probing Quantized Einstein-Rosen Waves with Massless Scalar Matter
The purpose of this paper is to discuss in detail the use of scalar matter
coupled to linearly polarized Einstein-Rosen waves as a probe to study quantum
gravity in the restricted setting provided by this symmetry reduction of
general relativity. We will obtain the relevant Hamiltonian and quantize it
with the techniques already used for the purely gravitational case. Finally we
will discuss the use of particle-like modes of the quantized fields to
operationally explore some of the features of quantum gravity within this
framework. Specifically we will study two-point functions, the Newton-Wigner
propagator, and radial wave functions for one-particle states.Comment: Accepted for publication in Physical Review
Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems
We study the Plancherel--Rotach asymptotics of four families of orthogonal
polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent
polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials
arise in indeterminate moment problems and three of them are birth and death
process polynomials with cubic or quartic rates. We employ a difference
equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to
a conjecture about large degree behavior of polynomials orthogonal with respect
to solutions of indeterminate moment problems.Comment: 34 pages, typos corrected and references update
Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal
In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a
resurgence representation for the gamma function, exploiting the reformulation
of the method of steepest descents by M. Berry and C. Howls (Berry and Howls,
Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was
able to derive a number of properties of the asymptotic expansion for the gamma
function, including explicit and realistic error bounds, the smooth transition
of the Stokes discontinuities, and asymptotics for the late coefficients. The
main aim of this paper is to modify the resurgence formula of Boyd making it
suitable for deriving better error estimates for the asymptotic expansions of
the gamma function and its reciprocal. We also prove the exponentially improved
versions of these expansions complete with error terms. Finally, we provide new
(formal) asymptotic expansions for the coefficients appearing in the asymptotic
series and compare their numerical efficacy with the results of earlier
authors.Comment: 22 pages, accepted for publication in Proceedings of the Royal
Society of Edinburgh, Section A: Mathematical and Physical Science
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