Representation of a class of multiple fractional part integrals and their closed form. Issue 7 (2nd July 2016)

• Record Type: Journal Article
• Authors: Li, Aijuan; Sun, Zhongfeng; Qin, Huizeng
• Is Part Of: Integral transforms and special functions. Volume 27:Issue 7(2016)
• Page Start: 578
• Page End: 591
• Publication Date: 2016-07-02
• Subjects: Multiple integrals -- fractional part -- Stieltjes constants -- Euler–Mascheroni constant -- Riemann Zeta function; 26A42 -- 26B15 -- 40A30; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2016.1167053
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: In this paper, the following generalized multiple fractional part integrals: I n, μ α 1, α 2, …, α n ( a ) = ∫ 0 1 ∫ 0 1 ⋯ ∫ 0 1 x 1 α 1 x 2 α 2 ⋯ x n α n 1 a x 1 x 2 ⋯ x n μ × d x 1 d x 2 ⋯ d x n, and k 1, k 2, …, k n I n, μ α 1, α 2, …, α n ( a ) = ∫ 0 1 ∫ 0 1 ⋯ ∫ 0 1 1 a x 1 x 2 ⋯ x n μ × ∏ i = 1 n x i α i ln k i ⁡ 1 x i d x 1 d x 2 ⋯ d x n, are considered for complex numbers μ, α i ( i = 1, 2, 3, …, n ), positive integers k i ( i = 1, 2, …, n ) and positive number a, where { u } denotes the fractional part of u . Furthermore, I n, μ α 1, α 2, …, α n ( a ) and k 1, k 2, …, k n I n, μ α 1, α 2, …, α n ( a ) can be transformed into the calculation of definite fractional part integral I μ α ( a ) and k I μ α ( a ), which overcomes the shortcomings of low speed and poor accuracy of the multiple case. Moreover, some identities and recursive formulas of the above integrals are obtained.