Related papers: Explicit evaluation of some integrals involving po…
We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the…
We present the evaluation of a family of exponential-logarithmic integrals. These have integrands of the form P(exp(x),ln(x)) where P is a polynomial. The examples presented here appear in sections 4.33, 4.34 and 4.35 in the classical table…
There are three kinds of multiple polylogarithms; complex, finite and symmetric. The dualities for the complex and finite cases are known. In this paper, we present proofs of them via iterated integrals and its symmetric counterpart by a…
We review a method for the algebraic treatment of a family of functions which contains the multiple polylogarithms, with applications to the symbolic calculation of Feynman integrals.
Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical…
Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind).…
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
We explore integrals of products of Legendre polynomials with a logarithmic weight function. More precisely, for Legendre polynomials $P_m$ and $P_n$ of orders $m$ and $n$, respectively, we provide simple derivations of the integrals $$\int…
This paper explores closed-form expressions for some polylogarithm integrals with integrands containing five parameters. These closed form expressions are given in terms of the Lerch transcendent function, which reduces, in some cases, to…
The polylogarithm function is one of the constellation of important mathematical functions. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics.…
We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2$. We obtain results…
In the present paper we extend the multiplicative integral to complex-valued functions of complex variable. The main difficulty in this way, that is the multi-valued nature of the complex logarithm, is avoided by division of the interval of…
We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and,…
We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,...,n_d}(z_1,...,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of…
In this paper, we establish some new integral inequalities for for m- and (alpha,m)-logarithmically convex functions.
We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved…
The fractional polylogarithms, depending on a complex parameter $\a$, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional…
Polylogrithmic functions, such as the logarithm or dilogarithm, satisfy a number of algebraic identities. For the logarithm, all the identities follow from the product rule. For the dilogarithm and higher-weight classical polylogarithms,…
In this talk, we discuss recent progress in the application of generalizations of polylogarithms in the symbolic computation of multi-loop integrals. We briefly review the Maple program MPL which supports a certain approach for the…
We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta…