相关论文: Computation of a Definite Integral
In this paper, we develop efficient and accurate algorithms for evaluating $\varphi(A)$ and $\varphi(A)b$, where $A$ is an $N\times N$ matrix, $b$ is an $N$ dimensional vector and $\varphi$ is the function defined by…
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent…
Formulas of $\pi(x)$-fine structure are presented.
In this paper we present a method to derive Pi(x) and other Arithemtical functions that can be generated by a Dirichlet series by variational principles,we use a variational method to determine the solution for a Fredholm integral equation…
We prove that $\alpha$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $\alpha \in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(\Delta x^{{1}/{8}}+\Delta…
The paper proposes an approximate expression for calculating very complex one-dimensional integrals depending on the parameter $a$. These integrals often occur in computational problems theory of magnetic solitons. The resulting analytical…
We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we…
In this paper, we study Pizzetti-type formulas for Stiefel manifolds and Cauchy-type formulas for the tangential Dirac operator from a distributional perspective. First we illustrate a general distributional method for integration over…
Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^T\Delta^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for…
For diophantine equations of the form ax^2+bxy+cy^2+g=0 over Z whose coefficients satisfy some assumptions, we show that a condition with respect to Artin reciprocity map, which we call the Artin condition, is the only obstruction to the…
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection…
We examine the use of the Euler-Maclaurin formula and new derived uniform asymptotic expansions for the numerical evaluation of the Lerch transcendent $\Phi(z, s, a)$ for $z, s, a \in \mathbb{C}$ to arbitrary precision. A detailed analysis…
This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order…
The Moll-Arias de Reyna integral [1] $$\int_0^{\infty}\frac{dx}{(x^2+1)^{3/2}}\frac{1}{\sqrt{\varphi(x)+\sqrt{\varphi(x)}}}$$ $$\varphi(x)=1+\frac{4}{3}\left(\frac{x}{x^2+1}\right)^2$$ is generalised and several values are given.
We construct spectral decomposition of 1D Fokker - Planck differential operator. This reveal solution of Cauchy problem. We develop fundamental solution of Cauchy problem and compare it with one obtained by other means in our former work…
We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present…
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 the present paper we extend the concepts of multiplicative de- rivative and integral to complex-valued functions of complex variable. Some drawbacks, arising with these concepts in the real case, are explained satis- factorily.…
In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…
We give an explicit upper bound for non-principal Dirichlet $L$-functions on the line $s=1+it$. This result can be applied to improve the error in the zero-counting formulae for these functions.