New integral representations of the polylogarithm function
Апстракт
Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li-s(z). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li-s(z) for any complex z for which vertical bar z vertical bar LT 1. Two are valid for all complex s, whenever Re s GT 1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function zeta(2n+1), n is an element of N, follow directly as corollaries of these representations.
Кључне речи:
polylogarithms / integral representation / Riemanns zeta function / Bernoulli polynomialsИзвор:
Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences, 2007, 463, 2080, 897-905
DOI: 10.1098/rspa.2006.1794
ISSN: 1364-5021
WoS: 000244255800001
Scopus: 2-s2.0-36349034660
Колекције
Институција/група
VinčaTY - JOUR AU - Cvijović, Đurđe PY - 2007 UR - https://vinar.vin.bg.ac.rs/handle/123456789/3155 AB - Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li-s(z). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li-s(z) for any complex z for which vertical bar z vertical bar LT 1. Two are valid for all complex s, whenever Re s GT 1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function zeta(2n+1), n is an element of N, follow directly as corollaries of these representations. T2 - Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences T1 - New integral representations of the polylogarithm function VL - 463 IS - 2080 SP - 897 EP - 905 DO - 10.1098/rspa.2006.1794 ER -
@article{ author = "Cvijović, Đurđe", year = "2007", abstract = "Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li-s(z). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li-s(z) for any complex z for which vertical bar z vertical bar LT 1. Two are valid for all complex s, whenever Re s GT 1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function zeta(2n+1), n is an element of N, follow directly as corollaries of these representations.", journal = "Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences", title = "New integral representations of the polylogarithm function", volume = "463", number = "2080", pages = "897-905", doi = "10.1098/rspa.2006.1794" }
Cvijović, Đ.. (2007). New integral representations of the polylogarithm function. in Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences, 463(2080), 897-905. https://doi.org/10.1098/rspa.2006.1794
Cvijović Đ. New integral representations of the polylogarithm function. in Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences. 2007;463(2080):897-905. doi:10.1098/rspa.2006.1794 .
Cvijović, Đurđe, "New integral representations of the polylogarithm function" in Proceedings of the Royal Society. A: Mathematical, Physical and Engineering Sciences, 463, no. 2080 (2007):897-905, https://doi.org/10.1098/rspa.2006.1794 . .