Related papers: Computing some Principal Value integrals without R…
We provide a simple approach for the evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind the method is to reduce the inverse transform to the solution of an ordinary…
Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data…
In solving the differential equation for a non damped harmonic oscillator one meets, after subjecting the equation to a Fourier transformation, an integration in the complex $\omega$ plane. In most cases such an integral is evaluated by…
These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have…
We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms…
We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal…
We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The…
Multi-head attention empowers the recent success of transformers, the state-of-the-art models that have achieved remarkable success in sequence modeling and beyond. These attention mechanisms compute the pairwise dot products between the…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are…
Usually such area of mathematics as differential equations acts as a consumer of results given by functional analysis. This article will give an example of the reverse interaction of these two fields of knowledge. Namely, the derivation and…
In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…
Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of…
In the introduction part of this paper, first of all, the concept of absolute integral sum of complex function is defined, as more general one with respect to the concept of integral as well as of integral sum of "ordinary'' integral…
The standard way of evaluating residues and some real integrals through the residue theorem (Cauchy's theorem) is well-known and widely applied in many branches of Physics. Herein we present an alternative technique based on the negative…
In this study, new master theorems and general formulas of integrals are presented and implemented to solve some complicated applications in different fields of science. The proposed theorems are considered to be generators of new problems,…
In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…
We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses,…
The explicit construction of direct and inverse Fourier's vector transform with discontinuous coefficients is presented. The technique of applying Fourier's vector transform with discontinuous coefficients for solving problems of…
These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle,…