Formulae concerning the computation of the Clausen integral Cl2(Θ)

Closed-form summation of some trigonometric series

“…First, it is of interest to check the validity of (8) and (9) (1 lb) C(*/3) = 6-"(l -2")f(i/) + (l/2)6-"(f(«/, 1/6) + C(v, 5/6)) The expression for Sv(n/6) obtained by using (9a) is somewhat more complicated, but it is not difficult to verify that it leads to the much simpler result It is obvious that the expressions in (7), (8) (7) is in fact a polylogarithm of order n . We note that our particular case is identical with that obtained by Lewin [9] which are given in [6] by observing that there exists the relationship [2] S2(a) = Cl2(a)-±Cl2(2a).…”

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.…”

SOME INTEGRAL REPRESENTATIONS OF THE CLAUSEN FUNCTION Cl2(x) AND THE CATALAN CONSTANT G

“…This integral was first treated by Clausen in 1832 [4] and has since then been investigated by many authors (see, e.g., [5], [7], [8], [9,Chapter 4], [12,Section 2.4], and many of the references cited therein). Some known properties and special values of the Clausen integral (or the Clausen function) include the periodic properties given by…”