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Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix--Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth…
Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…
For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth…
In this paper, we investigate the asymptotic properties of the generalised trigonometric integral $\operatorname{ti}(a, z, \alpha)$ and its associated modulus and phase functions for large complex values of $z$. We derive asymptotic…
We study the asymptotic error arising when approximating the Green's function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green's function of a regular Sturm-Liouville problem and for the…
In the book [Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Amer. Math. Soc., Providence, RI, 1997], a key and central ``algorithm'' was established, by which the coefficients of…
We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$ {\mathcal I}(t) = \int_{\mathbb R}({\rm e}^{it\phi(x)}-1) {\rm d} \mu(x) $$ where $\mu$ is a probability measure on…
Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic…
We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter $\Lambda$: $x^my"-\Lambda^2y=g(x)y$, with $m\in\mathbb{Z}$ and $g$…
The real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in…
The system of equations for parametric sub-resonant growth of the amplitude of oscillations was obtained. The time of turning point from the growing of the amplitude to the bounded oscillations in the slow variable was found. The comparison…
Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables.…
We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and…
We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The model considered in the paper is very general as we do not impose any…
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…
The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function…
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…
Asymptotics for Dickman's number theoretic function $\rho(u)$, as $u \rightarrow \infty$, were given de Bruijn and Alladi, and later in sharper form by Hildebrand and Tenenbaum. The perspective in these works is that of analytic number…
In recent years post-Newtonian approximations for isolated slowly-moving systems in general relativity have been studied by means of matched asymptotic expansions. A paper by Poujade & Blanchet in 2002 made great progress by effectively…
The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…