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We propose, analyze, and implement a quadrature method for evaluating integrals of the form $\int_0^2 f(s)\exp(zs)\, {\rm d}s$, where $z$ is a complex number with a possibly large negative real part. The integrand may exhibit exponential…
This article is the second of a series of three presenting an alternative method to compute the one-loop scalar integrals. It extends the results of the first article to general complex masses. Let us remind the main features enjoyed by…
The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…
We initiate a systematic framework for the analysis of analytic properties of finite Feynman integrals that are multiple polylogarithms. Based on the Feynman parameter representation in complex projective space, we make a complete…
In this paper, we focus on the explicit expression of an extended version of Riemann zeta function. We use two different methods, Mellin inversion formula and Cauchy's residue theorem, to calculate a Mellin-Barnes type integral of the…
In the theory of time scales, given $\mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration…
The generalized Stieltjes constants $\gamma\_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm…
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$…
We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating…
We use Poisson summation formula to calculate integrals of producs of sinc functions (cf. [4]) and related integrals as in [5] and [3]. We also generalize the one in [5] and introduce other remarkable integrals. Finally we give a sum…
We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems…
In this article we present logarithmic methods for solving first order and second order ordinary differential equations. The essence of the method is that we apply the basic properties derivatives and logarithms to reduce the number of…
The definite integral with the kernel x/(x^2+b^2)/[\exp(2\pi x)-1] integrated from x=0 to infinity is the main term of a representation of the Digamma-Function psi(b), the derivative of the logarithm of the Gamma-Function. We present…
We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite decomposition. We show that the terms in the…
For sampling from a log-concave density, we study implicit integrators resulting from $\theta$-method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the…
We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.
We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair…
The main aim is to obtain integral formulas for DAHA coinvariants and the corresponding inner products for any values of the DAHA parameters. In the compact case, our approach is similar to the procedure of ``picking up residues" due to…
As an application of Cauchy's Theorem we prove that $\int_0^1\arctan\left({\arctanh x-\arctan x\over \pi+\arctanh x-\arctan x}\right) {dx\over x}= {\pi\over 8}\log{\pi^2\over 8}$ answering a question first posted in Mathematics Stack…
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for theevaluation of Feynman diagrams. We describe the operational rules and illustrate the method…