Related papers: Asymptotics of some generalised sine-integrals

In our recent publications we have introduced the incomplete cosine expansion of the sinc function for efficient application in sampling [Abrarov & Quine, Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7 (2)…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and…

We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extend the…

In this paper, we apply the saddle-point method in conjunction with the theory of the N$\ddot{o}$rlund-Rice integrals to derive a precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for…

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed,…

Let $j \ge1$, $k\ge 0$ be real numbers and $\varphi(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum_{n\le x}\frac{\varphi\left ( \left [ \frac{x}{n} \right ]^{j}…

We derive two distinct asymptotic expansions for the zeros $j_{\nu,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_\nu^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $\nu$, for which we…

This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known $C_\infty$ "bump" function $\exp[-(1-x^2)^{-1}]$, deriving both the asymptotic decay rate $k^{-3/4} \exp(-\sqrt…

In this short lecture, we compute asymptotics of orthogonal polynomials, from a saddle point approximation. This is an example of a calculation which shows the link between integrability, algebraic geometry and random matrices.

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

In this study, we discuss the convergence and divergence of generalized integrals,\int_{0}^{+\infty}\frac{sin^{b}x}{x^{a}}dx(a\epsilon R^{+},b\epsilon N^{+}), and use the transformation method, the partial integration method, the…

Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion…

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain…

We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n!},\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the…

The asymptotic properties of integral operators with the generalized sine kernel acting on the real axis are studied. The formulas for the resolvent and the Fredholm determinant are obtained in the large x limit. Some applications of the…

In this paper, we first generalize the Fresnel integrals by changing of a path for integration in the proof of the Fresnel integrals by Cauchy's integral theorem. Next, according to oscillatory integral, we also obtain further…

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the…