School of Arts, Media and Computer Games
http://hdl.handle.net/10373/313
2016-02-11T02:35:27ZAsymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities
http://hdl.handle.net/10373/2240
Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities
Paris, Richard B.
We obtain an asymptotic evaluation of the integral ʃ ∞ √2n+1 e−x2 H2 n (x) dx for n → ∞, where Hn(x) is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the nth energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion.
2015-04-27T00:00:00ZThe asymptotic expansion of a generalisation of the Euler-Jacobi series
http://hdl.handle.net/10373/2229
The asymptotic expansion of a generalisation of the Euler-Jacobi series
Paris, Richard B.
We consider the asymptotic expansion of the sum
Sp(a; w) = ∞Ʃ n=1 e−anp/nw
as a → 0 in | arg a| <1/2π for arbitrary finite p > 0 and w > 0. Our attention is concentrated mainly on the case when p and w are both even integers, where the expansion consists of a finite algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for Sp (a; w) analogous to the well-known Poisson-Jacobi transformation for the sum with p = 2 and w = 0. Numerical results are given to illustrate the accuracy of the expansion obtained.
2016-01-01T00:00:00ZThe first millimeter detection of a non-accreting ultracool dwarf
http://hdl.handle.net/10373/2228
The first millimeter detection of a non-accreting ultracool dwarf
Williams, P. K. G.; Casewell, S. L.; Stark, Craig R.; Littlefair, S. P.; Helling, Ch.; Berger, E.
The well-studied M9 dwarf TVLM 513–46546 is a rapid rotator (P rot ~ 2 hr) hosting a stable, dipolar magnetic field of ~3 kG surface strength. Here we report its detection with ALMA at 95 GHz at a mean flux density of 56 ± 12 μJy, making it the first ultracool dwarf detected in the millimeter band, excluding young, disk-bearing objects. We also report flux density measurements from unpublished archival VLA data and new optical monitoring data from the Liverpool Telescope. The ALMA data are consistent with a power-law radio spectrum that extends continuously between centimeter and millimeter wavelengths. We argue that the emission is due to the synchrotron process, excluding thermal, free–free, and electron cyclotron maser emission as possible sources. During the interval of the ALMA observation that phases with the maximum of the object's optical variability, the flux density is higher at a ~1.8σ significance level. These early results show how ALMA opens a new window for studying the magnetic activity of ultracool dwarfs, particularly shedding light on the particle acceleration mechanism operating in their immediate surroundings.
2015-12-09T00:00:00ZOn the asymptotics of products related to generalizations of the Wilf and Mortini problems
http://hdl.handle.net/10373/2227
On the asymptotics of products related to generalizations of the Wilf and Mortini problems
Chen, Chao-Ping; Paris, Richard B.
In this paper, we present the connection between generalized Wilf’sand Mortini’s problems. We also consider the connection betweenanother infinite product formula and limit relation with severalparameters.
2015-12-11T00:00:00Z