Related papers: Asymptotics of some generalised sine-integrals

We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{\Gamma(1+\nu)|J_\nu(x)|}{(x/2)^\nu}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_\nu(x)$ is replaced by the…

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of…

Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion…

Using the saddle-point method, we compute an asymptotic, as $y \rightarrow \infty$, for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$…

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given…

We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with…

Perron's saddle-point method gives a way to find the complete asymptotic expansion of certain integrals that depend on a parameter going to infinity. We give two proofs of the key result. The first is a reworking of Perron's original proof,…

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \[\frac{2}{\pi}\int_0^{\pi/2} \!\!\!\begin{array}{c} \cos\\\sin\end{array}\!(x\tan u-\nu u)\,du\] are obtained for large real $x$ and large…

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found…

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, where $n$ is a…

We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies…

We consider saddle point integrals in d variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is…

Asymptotic expansions are obtained for contour integrals of the form \[ \int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt, \] in which $z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are analytic functions…

We consider the Pearcey integral $P(x,y)$ for large values of $\vert y\vert$ and bounded values of $\vert x\vert$. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is…

We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the…

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind…

We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0F_2$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F_2(-\!\!\!-;m+1, n+1; -(x/3)^3)\] as $x\to+\infty$, where…

We study the classical problem of finding asymptotics for the Bessel functions $J_{\nu}(z)$ and $Y_{\nu}(z)$ as the argument $z$ and the order $\nu$ approach infinity. We use blow-up analysis to find asymptotics for the modulus and phase of…

Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…

We consider the asymptotic expansion for $x\to\pm\infty$ of the entire function \[F_{n,\sigma}(x;\mu)=\sum_{k=0}^\infty \frac{\sin\,(n\gamma_k)}{\sin \gamma_k}\,\frac{x^k}{k! \Gamma(\mu-\sigma k)},\quad \gamma_k=\frac{(k+1)\pi}{2n}\] for…