Related papers: Landen transformations and the integration of rati…
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation…
In the present paper authors introduce the L_n-integral transform and the inverse integral transform for n = 2^k, k=0,1,2,..., as a generalization of the classical Laplace transform and the inverse Laplace transform, respectively.…
In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can…
We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors…
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…
In this article the algorithm for transformation of logic functions which are given by truth tables is considered. The suggested algorithm allows the transformation of many-valued logic functions with the required number of variables and…
In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though…
A two-parameters family of Backlund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows and are a time discretization of each of…
We propose a novel method for reconstructing Laurent expansion of rational functions using $p$-adic numbers. By evaluating the rational functions in $p$-adic fields rather than finite fields, it is possible to probe the expansion…
The Mellin transform of quartic products of shifted Airy functions is evaluated in a closed form. Some particular cases expressed in terms of the logarithm function and complete elliptic integrals special values are presented.
We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables. It also applies to systems of linear ODEs. It is…
We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…
We use some general properties, presented in previous work, to evaluate special cases of integrals relating Rogers-Ramanujan continued fraction, eta function and elliptic integrals.
In this work we derive results concerning Elliptic Functions using as tools general formulas from previus work.
We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been…