Representation of a class of multiple fractional part integrals and their closed form. Issue 7 (2nd July 2016) - Li, Aijuan; Sun, Zhongfeng; Qin, Huizeng
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.