Journal of Computational and Applied Mathematics, ISSN 0377-0427, 02/2019, Volume 347, pp. 330 - 342

In this paper, we investigate the integral of xnlogp(sin(x)) for natural numbers n and p. In doing so, we recover some well-known results and remark on some...

Riemann zeta function | Bell polynomial | Binomial coefficients | Combinatorics | Harmonic numbers | Log-sine integral | MATHEMATICS, APPLIED | Mathematics - Number Theory

Riemann zeta function | Bell polynomial | Binomial coefficients | Combinatorics | Harmonic numbers | Log-sine integral | MATHEMATICS, APPLIED | Mathematics - Number Theory

Journal Article

ACTA MATHEMATICA HUNGARICA, ISSN 0236-5294, 02/2020, Volume 160, Issue 1, pp. 45 - 57

The log-sine-polylog integrals were introduced by J. M. Borwein and A. Straub during their studies on special values of the log-sine integrals. In this paper...

SERIES | Clausen function | EPSILON-EXPANSION | MULTIPLE | alternating Euler sum | 2-LOOP | MATHEMATICS | the multiple zeta value data mine | Borwein-Straub algorithm | log-sine-polylog integral | Euler sum | multiple zeta value | log-sine integral | VALUES | polylogarihm

SERIES | Clausen function | EPSILON-EXPANSION | MULTIPLE | alternating Euler sum | 2-LOOP | MATHEMATICS | the multiple zeta value data mine | Borwein-Straub algorithm | log-sine-polylog integral | Euler sum | multiple zeta value | log-sine integral | VALUES | polylogarihm

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 5/2018, Volume 46, Issue 1, pp. 161 - 171

In this paper, we study the sequences $$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm...

33B15 | Polylogarithm functions | Fourier Analysis | Functions of a Complex Variable | Field Theory and Polynomials | Log-sine integrals | Hypergeometric function | Mathematics | Number Theory | Combinatorics | Gamma function | MATHEMATICS | HURWITZ ZETA-FUNCTION | Information science

33B15 | Polylogarithm functions | Fourier Analysis | Functions of a Complex Variable | Field Theory and Polynomials | Log-sine integrals | Hypergeometric function | Mathematics | Number Theory | Combinatorics | Gamma function | MATHEMATICS | HURWITZ ZETA-FUNCTION | Information science

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 03/2011, Volume 363, Issue 3, pp. 1463 - 1485

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor's multiple sine functions. It is shown that all the integrals...

Integers | Mathematical integrals | Sine function | Log integral function | Mathematical functions | Polynomials | Mathematical relations | Logarithms | Number theory | Multiple sine function | Mordell-Tornheim zeta value | Clausen function | Multiple zeta value | Log sine integral | MATHEMATICS | multiple zeta value | multiple sine function | FUNCTIONAL RELATIONS | SUMS

Integers | Mathematical integrals | Sine function | Log integral function | Mathematical functions | Polynomials | Mathematical relations | Logarithms | Number theory | Multiple sine function | Mordell-Tornheim zeta value | Clausen function | Multiple zeta value | Log sine integral | MATHEMATICS | multiple zeta value | multiple sine function | FUNCTIONAL RELATIONS | SUMS

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 12/2009, Volume 282, Issue 12, pp. 1709 - 1723

The Gamma function and its n th logarithmic derivatives (that is, the polygamma or the psi‐functions) have found many interesting and useful applications in a...

Gauss summation theorem | convolutions of Rayleigh functions | Goldbach−Euler series | psi‐function | Euler sums | Riemann Zeta function | Gamma function | polygamma functions | series associated with the Zeta function | log‐sine integrals | Hurwitz Zeta function | Goldbach-Euler series | Log-sine integrals | Polygamma functions | psi-Function | Convolutions of Rayleigh functions | Series associated with the Zeta function | INFINITE SERIES | REPRESENTATIONS | NUMBERS | log-sine integrals | RIEMANN-ZETA | DIGAMMA FUNCTIONS | MATHEMATICS | ZETA-FUNCTION | psi-function

Gauss summation theorem | convolutions of Rayleigh functions | Goldbach−Euler series | psi‐function | Euler sums | Riemann Zeta function | Gamma function | polygamma functions | series associated with the Zeta function | log‐sine integrals | Hurwitz Zeta function | Goldbach-Euler series | Log-sine integrals | Polygamma functions | psi-Function | Convolutions of Rayleigh functions | Series associated with the Zeta function | INFINITE SERIES | REPRESENTATIONS | NUMBERS | log-sine integrals | RIEMANN-ZETA | DIGAMMA FUNCTIONS | MATHEMATICS | ZETA-FUNCTION | psi-function

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 01/2004, Volume 147, Issue 3, pp. 645 - 667

In this paper we analysed some series involving ( ) k and ( ) k and some other related series and derived the integral representations of those series...

Integral representations | Catalan constant | Polylogarithms | Log-sine integrals | Riemann zeta function | Special functions

Integral representations | Catalan constant | Polylogarithms | Log-sine integrals | Riemann zeta function | Special functions

Journal Article

Integral Transforms and Special Functions, ISSN 1065-2469, 10/2011, Volume 22, Issue 10, pp. 767 - 783

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine integrals have been evaluated, in...

Psi (or Digamma) function | 33B15 | Bernoulli numbers | Hurwitz (or generalized) Zeta function | F | 11M35 | Gauss summation theorem for | Gamma function | log-sine integrals | Fourier series | Secondary 11B68 | Stirling numbers of the first kind | 1 | 2 | log-cosine integrals | 11B73 | Polygamma functions | 11M36 | 33B30 | Primary 11M06 | Log-sine integrals | Hurwitz (or generalized) zeta function | Log-cosine integrals | Psi (or digamma) function | MATHEMATICS, APPLIED | SERIES | SPECIAL VALUES | SUMS | POLYLOGARITHMS | MATHEMATICS | Gauss summation theorem for F-2 | DEFINITE INTEGRALS | RIEMANN ZETA-FUNCTION | Studies | Integrals

Psi (or Digamma) function | 33B15 | Bernoulli numbers | Hurwitz (or generalized) Zeta function | F | 11M35 | Gauss summation theorem for | Gamma function | log-sine integrals | Fourier series | Secondary 11B68 | Stirling numbers of the first kind | 1 | 2 | log-cosine integrals | 11B73 | Polygamma functions | 11M36 | 33B30 | Primary 11M06 | Log-sine integrals | Hurwitz (or generalized) zeta function | Log-cosine integrals | Psi (or digamma) function | MATHEMATICS, APPLIED | SERIES | SPECIAL VALUES | SUMS | POLYLOGARITHMS | MATHEMATICS | Gauss summation theorem for F-2 | DEFINITE INTEGRALS | RIEMANN ZETA-FUNCTION | Studies | Integrals

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2004, Volume 147, Issue 3, pp. 645 - 667

In this paper we analysed some series involving 2k k −1k −n and 2k k −2k −n and some other related series and derived the integral representations of those...

Log-sine integrals | Riemann zeta function | Special functions | Integral representations | Catalan constant | Polylogarithms

Log-sine integrals | Riemann zeta function | Special functions | Integral representations | Catalan constant | Polylogarithms

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 05/2015, Volume 193, pp. 74 - 88

Polylogarithms appear in many diverse fields of mathematics. Herein, we investigate relations amongst the restricted class of Nielsen-type (essentially, height...

Multiple zeta values | Log-sine integrals | Clausen functions | Multiple polylogarithms | MATHEMATICS | VALUES | 3-LOOP FEYNMAN DIAGRAMS

Multiple zeta values | Log-sine integrals | Clausen functions | Multiple polylogarithms | MATHEMATICS | VALUES | 3-LOOP FEYNMAN DIAGRAMS

Journal Article

Journal of Number Theory, ISSN 0022-314X, 05/2015, Volume 150, pp. 98 - 119

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of...

Harmonic sums | Binomial transformations | Mellin transforms | Bell polynomials | Polylogarithms | Log-sine integrals | Zeta values | MATHEMATICS | Poly logarithms | SUMS | Mathematics | Number Theory

Harmonic sums | Binomial transformations | Mellin transforms | Bell polynomials | Polylogarithms | Log-sine integrals | Zeta values | MATHEMATICS | Poly logarithms | SUMS | Mathematics | Number Theory

Journal Article

Proceedings of the American Mathematical Society, ISSN 0002-9939, 09/2015, Volume 143, Issue 9, pp. 3743 - 3752

of the Riemann zeta function is rich enough to capture real numbers in an approximation aspect. Precisely, we prove that any real number can be strongly...

Riemann zeta function | Approximation property | Special values | Even values | Odd values | approximation property | MATHEMATICS, APPLIED | SERIES | REPRESENTATIONS | APPROXIMATION | odd values | even values | LOG-SINE INTEGRALS | MATHEMATICS | RAMANUJAN | SPIRIT | special values

Riemann zeta function | Approximation property | Special values | Even values | Odd values | approximation property | MATHEMATICS, APPLIED | SERIES | REPRESENTATIONS | APPROXIMATION | odd values | even values | LOG-SINE INTEGRALS | MATHEMATICS | RAMANUJAN | SPIRIT | special values

Journal Article

Journal of the Australian Mathematical Society, ISSN 1446-7887, 02/2012, Volume 92, Issue 1, pp. 15 - 36

We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of...

Clausen functions | Mahler measure | log-sine integrals | multiple polylogarithms | multiple zeta values | INTEGRALS | MATHEMATICS | EPSILON-EXPANSION | VALUES | 2-LOOP | SUMS

Clausen functions | Mahler measure | log-sine integrals | multiple polylogarithms | multiple zeta values | INTEGRALS | MATHEMATICS | EPSILON-EXPANSION | VALUES | 2-LOOP | SUMS

Journal Article

Computer Physics Communications, ISSN 0010-4655, 10/2005, Volume 172, Issue 1, pp. 45 - 59

Generalized log-sine functions Lsj(k)(θ) appear in higher order ε-expansion of different Feynman diagrams. We present an algorithm for the numerical evaluation...

Feynman integrals | Polylogarithms | Generalized log-sine functions | SERIES | EPSILON-EXPANSION | SCALE DIAGRAMS | PHYSICS, MATHEMATICAL | 2-LOOP | INTEGRALS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FEYNMAN DIAGRAMS | polylogarithms | POLE-MASS | FORMULAS | generalized log-sine functions | RENORMALIZATION

Feynman integrals | Polylogarithms | Generalized log-sine functions | SERIES | EPSILON-EXPANSION | SCALE DIAGRAMS | PHYSICS, MATHEMATICAL | 2-LOOP | INTEGRALS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FEYNMAN DIAGRAMS | polylogarithms | POLE-MASS | FORMULAS | generalized log-sine functions | RENORMALIZATION

Journal Article

Proceedings of the 36th international symposium on symbolic and algebraic computation, 06/2011, pp. 43 - 50

We study generalized log-sine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at...

log-sine integrals | clausen functions | multiple polylogarithms | multiple zeta values

log-sine integrals | clausen functions | multiple polylogarithms | multiple zeta values

Conference Proceeding

Integral Transforms and Special Functions, ISSN 1065-2469, 01/2008, Volume 19, Issue 1, pp. 65 - 79

The main object of this paper is to introduce and investigate a mutiple Hurwitz-Lerch Zeta function Phi(n)(z, s, a), which generalizes the Hurwitz-Lerch Zeta...

Double Gamma function | Polylogarithms | Log-sine integrals | Riemann Zeta function | Gamma function | Series associated with the Zeta function | Hurwitz Zeta function | MATHEMATICS, APPLIED | SERIES | DETERMINANTS | gamma function | log-sine integrals | SUMS | POLYNOMIALS | MATHEMATICS | FAMILIES | APOSTOL-BERNOULLI | MULTIPLE GAMMA-FUNCTIONS | double gamma function | series associated with the Zeta function | polylogarithms

Double Gamma function | Polylogarithms | Log-sine integrals | Riemann Zeta function | Gamma function | Series associated with the Zeta function | Hurwitz Zeta function | MATHEMATICS, APPLIED | SERIES | DETERMINANTS | gamma function | log-sine integrals | SUMS | POLYNOMIALS | MATHEMATICS | FAMILIES | APOSTOL-BERNOULLI | MULTIPLE GAMMA-FUNCTIONS | double gamma function | series associated with the Zeta function | polylogarithms

Journal Article

Integers, ISSN 1867-0652, 12/2012, Volume 12, Issue 6, pp. 1179 - 1212

We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in “Log-sine evaluations of Mahler measures” and “Special...

Multiple Polylogarithms | Multiple Zeta Values | Log-Sine Integrals | Mahler Measure | Clausen Functions

Multiple Polylogarithms | Multiple Zeta Values | Log-Sine Integrals | Mahler Measure | Clausen Functions

Journal Article

17.
Full Text
Mahler measures, short walks and log-sine integrals

: a case study in hybrid computation

Proceedings of the 2011 International Workshop on symbolic-numeric computation, 06/2012, pp. 1 - 1

The Mahler measure of a polynomial of several variables has been a subject of much study over the past thirty years. Very few closed forms are proven but many...

Mahler measures | log-sine integral | short walks

Mahler measures | log-sine integral | short walks

Conference Proceeding

APPLIED MATHEMATICS AND COMPUTATION, ISSN 0096-3003, 01/2004, Volume 147, Issue 3, pp. 645 - 667

In this paper we analysed some series involving ((2k)(k))(-1) k(-n) and ((2k)(k))(-2) k(-n) and some other related series and derived the integral...

MATHEMATICS, APPLIED | catalan constant | Riemann zeta function | integral representations | polylogarithms | log-sine integrals | special functions

MATHEMATICS, APPLIED | catalan constant | Riemann zeta function | integral representations | polylogarithms | log-sine integrals | special functions

Journal Article

Computer Physics Communications, ISSN 0010-4655, 2005, Volume 172, Issue 1, pp. 45 - 59

Generalized log-sine functions Ls j ( k ) ( θ ) appear in higher order ɛ-expansion of different Feynman diagrams. We present an algorithm for the numerical...

Feynman integrals | Polylogarithms | Generalized log-sine functions | Analysis | Libraries | Algorithms

Feynman integrals | Polylogarithms | Generalized log-sine functions | Analysis | Libraries | Algorithms

Journal Article

Experimental Mathematics, ISSN 1058-6458, 01/2001, Volume 10, Issue 1, pp. 25 - 34

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums)....

Clausen's function | multiple zeta values | multiple Clausen values | binomial sums | log-sine integrals | polylogarithms | Apery sums | Multiple Clausen values | Apéry sums | Polylogarithms | Log-sine integrals | Multiple zeta values | Binomial sums | MATHEMATICS | 05Axx | 11Mxx | 33Bxx | 11Bxx

Clausen's function | multiple zeta values | multiple Clausen values | binomial sums | log-sine integrals | polylogarithms | Apery sums | Multiple Clausen values | Apéry sums | Polylogarithms | Log-sine integrals | Multiple zeta values | Binomial sums | MATHEMATICS | 05Axx | 11Mxx | 33Bxx | 11Bxx

Journal Article

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