Related papers: Asymptotics of some generalised sine-integrals
We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…
We find asymptotical expansions as $\nu \to 0$ for integrals of the form $\int_{\mathbb{R}^d} F(x) / \big(\omega(x)^2 + \nu^2\big)\, dx$, where sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions for their behaviour…
We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_\nu(an^px)}{(an^px/2)^\nu}\qquad (x>0),\] in the limit $a\to0$, where…
In this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the L\'evy…
Consider an integral $I(s):=\int_0^T e^{-s(x^2-icx)}dx$, where $c>0$ and $T>0$ are arbitrary positive constants. It is proved that $I(s)\sim \frac{i}{sc}$ as $s\to +\infty$. The asymptotic behavior of the integral…
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical…
In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N…
The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal…
We show how to determine the asymptotics of a certain Selberg-type integral by means of tools available in the theory of (generalised) hypergeometric series. This provides an alternative derivation of a result of Carr\'e, Deneufch\^atel,…
In this paper, we prove asymptotic expansions of generalized partial theta functions with a nonprincipal Dirichlet character and relate these expansions to certain $L$-series.
The aim of this work is to analyze general infinite sums containing modified Bessel functions of the second kind. In particular we present a method for the construction of a proper asymptotic expansion for such series valid when one of the…
We generalize the Mittag-Leffler function by attaching an exponent to its Taylor coefficients. The main result is an asymptotic formula valid in sectors of the complex plane, which extends work by Le Roy [Bull. des sciences math. 24, 1900]…
The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem - too much freedom! The SCE can be used to generate an enormous number…
Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…
A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\cap\Lambda_n}g_n(x)e^{-nh_n(x)} have similar…
The use of hyperasymptotics and the Weniger transformation has been proposed, in a joint fashion, for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle-point integrals {evaluated across…
We obtain exponential moment asymptotics for the Bessel point process. As a direct consequence, we improve on the asymptotics for the expectation and variance of the associated counting function, and establish several central limit…
We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying…
We obtain the rigorous uniform asymptotics of a particular integral where a stationary point is close to an endpoint. There exists a general method introduced by Bleistein for obtaining uniform asymptotics in this situation. However, this…
This paper develops asymptotic theory of integrals of empirical quantile functions with respect to random weight functions, which is an extension of classical $L$-statistics. They appear when sample trimming or Winsorization is applied to…