相关论文: Numerical Evaluation Of the Oscillatory Integral o…
Loop calculations involve the evaluation of divergent integrals. Usually [1] one computes them in a number of dimensions different than four where the integral is convergent and then one performs the analytical continuation and considers…
In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is…
A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[…
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are…
We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the…
We present methods for the numerical evaluation of the master integrals that appear in the calculation of scattering amplitudes at higher order in perturbative quantum field theory. We follow the general strategy of solving first-order…
Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the…
We calculate the two-particle irreducible (2PI) effective potential of the O(N) linear sigma model in 1+1 dimensions. The approximations we use are the next-to-leading order of a 1/N expansion (for arbitrary N) and a kind of "resummed loop…
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 provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and…
In this paper, we propose a numerical method of computing an Hadamard finite-part integral, a finite value assigned to a divergent integral, with a non-integral power singularity at the endpoint on a half infinite interval. In the proposed…
We apply a result of David and Jon Borwein to evaluate a sequence of highly-oscillatory integrals whose integrands are the products of a rapidly growing number of sinc functions. The value of each integral is given in the form $\pi(1-t)/2$,…
We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…
In this paper, we propose a numerical method of computing Hadamard finite-part integrals with an integral power singularity at the endpoint on a half infinite interval, that is, a finite value assigned to a divergent integral with an…
The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier integral. The double exponential transformation is not only useful for numerical computations but it is…
We present a new method for the numerical evaluation of arbitrary loop integrals in dimensional regularization. We first derive Mellin-Barnes integral representations and apply an algorithmic technique, based on the Cauchy theorem, to…
In this paper, we propose a numerical method of computing a Hadamard finite-part integral with a non-integral power singularity at an endpoint, that is, a finite part of a divergent integral as a limiting procedure. In the proposed method,…
We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower…
A general framework of Numerical Singular Integrals (NSI) method based on the Integration By Parts (IBP) has been developed for integrals involving singular and nearly singular integrands, or NSI-IBP. Through a general integration by parts…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…