An efficient calculation of the Clausen functions Cl n (θ)(n≥2)

Abstract

The Clausen functions appear in many problems, such as in the computation of singular integrals, quantum field theory, and so on. In this paper, we consider the Clausen functions Cl n (θ ) with n ≥ 2. An efficient algorithm for evaluating them is suggested and the corresponding convergence analysis is established. Finally, some numerical examples are presented to show the efficiency of our algorithm.

Approximations for Apery's constant $ζ(3)$ and rational series representations involving $ζ(2n)$

“…It is well-known that ζ(n) − 1 converges to zero rapidly for large values of n. In [30], Wu, Zhang and Liu derive the following representation for the Clausen function Cl 2 (θ), (11) Cl…”

“…This integral was considered for the first time by Clausen in 1832 ( [15]), and it was investigated later by many authors [9,10,11,12,13,14,22,23,29,30]. The Clausen functions are very important in mathematical physics.…”

Numerical solution of a certain hypersingular integral equation of the first kind

“…with s = x k−1 , where the identity (see [3]) is the Clausen's integral [33]. Since the integrand function in (4.21) is continuous except one removable discontinuity at s, by the error estimate of the midpoint for Riemann integrals, we have…”

Generalized rational zeta series for ζ(2n) and ζ(2n+1)

“…The Clausen function also has a power series representation which will be used later in the paper. It is given as There are also higher order Clausen-type function defined as (4) Cl The Clausen function is widely studied and has many applications in mathematics and mathematical physics ( [5], [6], [9], [12], [14], [15], [17]). We will also discuss the Dirichlet beta function is known as Catalan's constant.…”

“…The Clausen function is widely studied and has many applications in mathematics and mathematical physics ( [5], [6], [9], [12], [14], [15], [17]). We will also discuss the Dirichlet beta function…”