Related papers: On Non-Oscillating Integrals for Computing Inhomog…
We study stochastic convolutions providing by fundamental solutions of a class of integrodifferential equations which interpolate the heat and the wave equations. We give sufficient condition for the existence of function--valued…
Using a differential equation approach asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The…
We present some observations on the distribution of the zeros of solutions of the nonhomogeneous Airy's equation. We show the existence of a principal family of solutions, with simple zeros, and particular solutions, characterized by a…
This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral…
Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from…
Asymptotic formulas are derived for the Stieltjes-Wigert polynomials $S_n(z;q)$ in the complex plane as the degree $n$ grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc…
The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary…
Over short time intervals planetary ephemerides have been traditionally represented in analytical form as finite sums of periodic terms or sums of Poisson terms that are periodic terms with polynomial amplitudes. Nevertheless, this…
A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly…
We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates…
Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas…
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…
A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a…
We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative…
This work develops an analytic framework for the study of the $\zeta$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $\zeta$-functions…
Green's function of the problem describing steady forward motion of bodies in an open ocean in the framework of the linear surface wave theory (the function is often referred to as Kelvin's wave source potential) is considered. Methods for…
We calculate the special values of the spectral zeta function of the non-commutative harmonic oscillator, and give a general formula for them as integrals of certain algebraic functions. This is a generalization of the result by…
The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We…