Trigonometric Integrals Research Articles

A new approach to MacMahon’s Partition Analysis and associated applications

We introduce a new elimination procedure for Partition Analysis to obtain generating functions free of Omega variables starting with a crude generating function containing or not factored polynomials. In contrast to Han’s approach (Ann Comb. 2003;7(4):467–480), we work with factored polynomials by judiciously constructing the quotient of a series modulo a certain prime ideal. Our results include diverse areas such as an Omega-based approach to compute the Hadamard product of power series and certain trigonometric integrals generalizing the main result of Kim (J Integer Seq. 2009;12:09.7.4) and recovering an integral identity of Boros and Moll (J Comput Appl Math. 1999;106(2):361–368), respectively. Furthermore, we construct a new Omega representation of analytic matrix functions which requires no spectral information and uses the matrix rank to simplify the calculations. By specializing to tridiagonal matrices, the main result of Kiliç (Appl Math Comput. 2008;197(1):345–357) is obtained.

Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions

Abstract Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.

Low-regularity exponential-type integrators for the Zakharov system with rough data in all dimensions

We propose and analyze a type of low-regularity exponential-type integrators (LREIs) for the Zakharov system (ZS) with rough solutions. Our LREIs include a first-order integrator and a second-order one, and they achieve optimal convergence under weaker regularity assumptions on the exact solution compared to the existing numerical methods in literature. Specifically, the first-order integrator exhibits linear convergence in H m + 2 ( T d ) × H m + 1 ( T d ) × H m ( T d ) H^{m+2}(\mathbb {T}^d)\times H^{m+1}(\mathbb {T}^d)\times H^m(\mathbb {T}^d) for solutions in H m + 3 ( T d ) × H m + 2 ( T d ) × H m + 1 ( T d ) H^{m+3}(\mathbb {T}^d)\times H^{m+2}(\mathbb {T}^d)\times H^{m+1}(\mathbb {T}^d) if m > d / 2 m>d/2 , meaning that only the boundedness of one additional derivative of the solution is required to achieve the first-order convergence. While for the second-order integrator, we show that it achieves second-order accuracy by requiring the boundedness of two additional spatial derivatives of the solution. The order of additional derivatives required is reduced by half compared to the classical trigonometric integrators. The main techniques to design the integrators include a reformulation by introducing new variables to exclude the loss of spatial regularity in the original ZS, accurate integration for the dominant term in the linear part of the equations and appropriate approximations (or averaging approximations) to the exponential phase functions involving the nonlinear interactions. Numerical comparisons with classical integrators confirm that our newly proposed LREIs are superior in accuracy and robustness for handling rough data.

On the uniform convergence and integrability of special trigonometric integrals

On the uniform convergence and integrability of special trigonometric integrals

Positivity of oscillatory integrals and Hankel transforms

ABSTRACT In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm's theory. As applications, positivity criteria are obtained for Hankel transforms as well as trigonometric integrals defined on the positive real line.

The Siren Call of Calculus A Review of Steven Strogatz’s Infinite Powers: How Calculus Reveals the Secrets of the Universe

I hate calculus. Which is a lie: I secretly love it. I enjoy teaching it, drawing intricate diagrams of where the definition of the derivative comes from, stumping my students on convoluted trigonometric integrals with solutions dependent on partial fractions, hammering mnemonics for the derivative rules—all the joys of teaching calculus. And I enjoyed learning it, too. Well, not failing it the first time, but after that. I long for my undergraduate time spent in the reading room of Rutherford South re-arranging integrals à la Fubini’s Theorem, the massive lecture halls of anxious engineering students copying every stroke of chalk from the front, and the arithmetic errors costing me scores of marks from my tests. A calculus course presents the liturgy of undergraduate mathematics: students and instructors alike gather, willfully or otherwise, to engage in the ritualistic celebration of the mysteries of the infinite.

On the Calculation of Several Definite Integrals by Residue Theorem

The Cauchy’s residue theorem is one of the most important theorems in complex analysis at all times, and it is demonstrated that using the residue theorem is an easier and faster method to calculate some types of real improper integrals when the targeted integrals are hard or even impossible to deal with by conventional approaches. This paper considers several types of integrals including trigonometric integrals and integrals involving logarithmic function and power function. The general methods for calculating these integrals are presented and typical examples to illustrate how to use the methods are shown. The types of integrals in the paper are useful in many fields and have applications in the engineering and scientific research. In complex analysis, the residue theorem is a powerful tool for calculating the path integrals of analytic functions along closed curves, and can also be used to calculate the integrals of real functions.

Application and Analysis of The Residue Theorem

This paper studies the derivation of the residue theorem and its application. The residue theorem is an important part of the theory of complex functions and plays a critical role in the development and application of complex functions. At the same time, the residue theorem advances the method of solving the value of definite integrals to a new stage. The issue of the Cauchy residue theorem has received considerable attention. Also, residue theorem lays the foundation for the development of integral theory, and it has been widely used in many scientific areas. The goal of this study is to investigate the properties and applications of the residue theorem. To this end, the strategies to deduce the residue theorem are summarized. Then this paper uses several examples that relate to the residue theorem to help people understand the residue theorem more deeply. The integrals under consideration include trigonometric integrals and integrals involving power functions.

One-stage explicit trigonometric integrators for effectively solving quasilinear wave equations

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations.

Letter to the Editor: Addition to the Paper “On the Convergence Exponent of Trigonometric Integrals”

Letter to the Editor: Addition to the Paper “On the Convergence Exponent of Trigonometric Integrals”

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

Summation of trigonometric Fourier integrals by method of G.F. Voronoi

In this paper we establish sufficient conditions of summability, by functional method of G.F. Voronoi, of Fourier integral of a function $f(t) \in L_{(-\infty, \infty)}$.

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

Continuous trigonometric collocation polynomial approximations with geometric and superconvergence analysis for efficiently solving semi-linear highly oscillatory hyperbolic systems

In this paper, based on the continuous collocation polynomial approximations, we derive and analyse a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system. The symmetry, convergence and energy conservation of the continuous collocation polynomial approximations are rigorously analysed in details. Moreover, we also proved that the continuous collocation polynomial approximations could achieve at superconvergence by choosing suitable collocation points. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods in the literature.

Kesalahan Pemahaman Konsep Peserta Didik dalam Menyelesaikan Soal-Soal Integral Lipat Dua pada Koordinat Polar

Penelitian ini adalah penelitian deskriptif kualitatif untuk menganalisis kesalahan pemahaman konsep peserta didik dalam menyelesaikan soal-soal integral lipat dua pada koordinat polar. Tiga puluh peserta didik yang mengambil mata kuliah kalkulus peubah banyak menjadi subjek dalam penelitian ini. Data dikumpulkan dengan menggunakan tes dan dokumentasi. Hasil jawaban tes peserta didik serta dokumentasi hasil pengerjaan peserta didik, dikoreksi, dianalisis berdasarkan indikator pemahaman konsep dan didisplay data. Analisis kesalahan jawaban peserta didik berdasarkan indikator pemahaman konsep yakni (1) menyatakan ulang konsep integral. (2) merepresentasikan konsep ke bentuk lain, (3) memilih dan menggunakan prosedur serta (4) menggunakan konsep dan algoritma pemecahan masalah. Dalam menyatakan ulang konsep integral peserta didik langsung menggunakan batas atas dan batas bawah integrasi tanpa menuliskan terlebih dahulu hasil pengintegralan. Kesalahan merepresentasikan fungsi ke grafik adalah, fungsi langsung disketsakan tanpa membuat table, memplot titik dan membuat kurva. Kesalahan dalam memilih dan menggunakan prosedur dan penggunaan algoritma pemecahan masalah dikarenakan kurangnya pengetahuan awal peserta didik mengenai materi prasyarat yakni, integral trigonometri dan fungsi trigonometri.

On points of strong summability of trigonometric integrals

On points of strong summability of trigonometric integrals

Hypergeometric Representations and Differential-Difference Relations for Some Kernels Appearing in Mathematical Physics

We investigate the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations they satisfy. Mathematically, the results include calculation of certain trigonometric double integrals and derivation of their other properties. Furthermore, they represent a potentially useful tool in matter relaxation in an external field, the study of nanoelectronic electrolyte-based systems and dynamics of charge carriers in media with obstacles.

Down with Weierstrass!

We offer an alternative to the so-called Weierstrass substitution for a class of trigonometric integrals, with references to Euler, Hardy, and others.

Asymptotic preserving trigonometric integrators for the quantum Zakharov system

We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime $$\vartheta = 1$$ as well as in the classical regime $$\vartheta \rightarrow 0$$ without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit $$\vartheta \rightarrow 0$$ uniformly in the time discretization parameter. Numerical experiments underline the favorable error behavior of the new schemes with first- and second-order time convergence uniformly in $$\vartheta $$ , first-order asymptotic convergence in $$\vartheta $$ and long time structure preservation properties.

On Energy Conservation by Trigonometric Integrators in the Linear Case with Application to Wave Equations

On Energy Conservation by Trigonometric Integrators in the Linear Case with Application to Wave Equations