Related papers: Exact Evaluation Of Some Highly Oscillatory Integr…
In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin…
Two classes of infinite series involving harmonic numbers and the binomial coefficient $C(3n,n)$ are evaluated in closed form using integrals. Several remarkable integral values and difficult series identities are stated as special cases of…
We show that the use of generalized multivariable forms of Hermite polynomials provide an useful tool for the evaluation of families of elliptic type integrals often encountered in electrostatic and electrodynamics
We give a natural notion of (non-exact) integral functor in the context of k-linear and graded categories. In this broader sense, we prove that every k-linear and graded functor is integral.
Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…
A recipe is presented for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.
We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical…
A class of first order linear impulsive differential equation with continuous and piecewise constant arguments is studied. Sufficient conditions for the oscillation of the solutions are obtained.
In this paper, we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integrals is the linear perturbation of a function having $D$ type singularities. We consider estimates for the…
In this note, we presented a new decomposition of elements of finite fields of even order and illustrated that it is an effective tool in evaluation of some specific exponential sums over finite fields, the explicit value of some…
The general version of the bosonic harmonic oscillator realisation of bosonic q-oscillators is given. It is shown that the currently known realisation is a special case of our general solution. The investigation has been performed at the…
The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean…
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
This paper formalize the existence's proof of first-integrals for any second order ODE, allowing to discriminate periodic orbits. Up to the author's knowledge, such a powerful result is not available in the literature providing a tool to…
This is a conspectus of definite integrals, products and series. These formulae involve special functions in the integrand and summand functions and closed form solutions. Some of the special cases are stated in terms of fundamental…
We study the explicit formula of Euler numbers and polynomials of higher order
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
A new general formula for the number of conjugacy classes of subgroups of given index in a finitely generated group is obtained.
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.