Evaluating Log-Tangent Integrals via Euler Sums

. An investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.


Introduction, preliminaries and notation
In this paper we investigate the representations of integrals of the type in terms of special functions such as zeta functions, Dirichlet eta functions, beta functions and others.We note that δ = ±1, a and b are fixed real numbers and p ∈ N for the set of complex numbers C, natural numbers N, the set of real numbers R and the set of positive real numbers, R + .In fact, for δ = ±1, p ∈ N and b ∈ R + , the integral I (δ, a, b, p) converges if and only if a + b > −1; also the integral J (δ, a, b, p) converges if and only if a + b > −1 and a < 1.In the two papers [9] and [8] the authors investigate log-tangent integrals and the Riemann zeta function.In the paper [9], they showed that for any square integrable function f ∈ (0, π/2) the integral L (f ) := where c k (P ) = 1 − 2 −1−2k P (2k−1) (1) + P (2k−1) (0) and P (k) (x) denotes the k th derivative of P at the point x.In the second paper [8], the authors show that the integrals involving the log-tangent function can be evaluated by some series involving the harmonic numbers, H n .For certain square integrable functions f ∈ (0, π/2), then The authors also point out that integrals involving log-tangent functions have important applications in many fields of mathematics including evaluation of classical, semi classical and quantum entropies of position and momentum.We can see that (1.1) and (1.2) reduce to The following notation and results will be useful in the subsequent sections of this paper.The generalized p-order harmonic numbers, H is a special case of the Dirichlet beta function, see [1] where the Gamma function and with functional equation extending the Dirichlet Beta function to the left hand side of the complex plane Re(z) ≤ 0. For the odd case z = 2q + 1, q ∈ N 0 where E q are the Euler numbers, that is the integer numbers obtained as the coefficients of z q /q! in the Taylor series expansion of sech(z) , |z| < π/2.For the even case, utilizing the following from Kölbig [13] we have upon adding, where B n are the Bernoulli numbers, the rational coefficients of z q /q! in the Taylor expansion of z/ (e z − 1) , |z| < 2π.The Lerch transcendent, t is defined for |z| < 1 and Re (a) > 0 and satisfies the recurrence It is known that the Lerch transcendent extends by analytic continuation to a function Φ (z, t, a) which is defined for all complex t, z ∈ C − [1, ∞) and a > 0. The Lerch transcendent generalizes the Hurwitz zeta function at z = 1, (1.4) The Dirichlet lambda function, λ (p) is where is the alternating zeta function and η (1) = ln 2. We know that for n ≥ 1, ψ(n+ 1) − ψ(1) = H n with ψ(1) = −γ, where γ is the Euler Mascheroni constant and ψ(n) is the digamma function.For real values of x, ψ(x) is the digamma (or psi) function defined by leading to the telescoping sum: The polygamma function and has the recurrence The connection of the polygamma function with harmonic numbers is, and the multiplication formula is, see [25] for p a positive integer and δ p,k is the Kronecker delta.The following lemma will be useful.
A. Sofo By a similar argument we have The following Lemma deals with the representation of an alternating Euler sum and utilizes the result from Lemma 1.
Proof.From Knopp [12], using the definition of the Polylogarithm which is valid for t ∈ C and |z| < 1 and can be extended to |z| ≥ 1 by analytic continuation.We have, by series expansion and replacing z := −x 2 , we get Integrating the LHS by parts, for x ∈ (0, 1), or using a mathematical package, reduces to A shift in the counter n results in and making use of Lemma 1, we have that We expect that integrals of the type (1.1) and (1.2) may be represented by Euler sums and therefore in terms of special functions such as the Riemann zeta function.A search of the current literature has found some examples for the representation of the log tangent integral in terms of Euler sums, see [2,8,9] and [15].The following papers, [22,23] also examined some integrals in terms of Euler sums.Some other important sources of information on log-tangent integrals and Euler sums are the works in the excellent books [14,26] and [27].Other useful references related to the representation of Euler sums in terms of special functions include [3,4,6,17,18,19,22,24].Some examples will be given highlighting specific cases of the integrals, some of which are not amenable to a computer mathematical package.

Analysis of integrals
where are harmonic numbers of order p + 1.
A. Sofo Proof.Applying the Taylor series expansion we can write for x ∈ (0, 1) , by Fubini's theorem, see [5], we are assured convergence and upon reversing the order of integration and summation we have Applying the Leibnitz differentiation rule, we have From the relation (1.5) we have, and Theorem 1 follows.
The case b = 1 gives us the following interesting relation.
Corollary 1.Let the conditions of Theorem 1 hold then, for b = 1 Proof.First, by the substitution x = tan θ, It is known that the Cauchy product of two convergent series, see [7] n≥0 where c n = n j=0 a j b n−j .For x ∈ (−1, 1) then by a Taylor series expansion,

If we know match (2.2) with (2.1) we obtain (2.3).
There are some interesting special cases of Theorem 1 and we present mainly the case of a = 0, b = 1.Consider the following.
Proof.If we make the substitution x = tan θ, we obtain

A. Sofo
From from the multiplication formula (1.6), see [20], we know , therefore the simplified integral is apply partial fraction decomposition to the second sum and the corollary follows.
Remark 1. From Theorem 1, using the functional equation we can evaluate, for b > 0, I (1, a, −b, p) .Consider the case a = 1, b > 0, p ∈ N, then where Some illustrative examples of Theorem 1 and its corollaries follow.To evaluate the resultant Euler sums for these examples we need a mixture of identities, including (1.6), Lemma 1 and some from [21].
Example 1.For p = 1 here we require, from Lemma 2 where W (3) is given by (1.7), and in a similar fashion The degenerate case .

A. Sofo
Here we need the following result which can be evaluated by the method of Lemma 2: Now consider the integral (1.2).
where β (•) is the Dirichlet beta function (1.3) and I (1, 0, 1, 2q − 1) is obtained from (2.4). Proof.Consider we notice that f (p, x) is continuous, bounded and differentiable on the interval x ∈ (0, 1] , with lim where in the third integral we have made the transformation xy = 1, now use the relation arctan (x) + arctan 1 x = π 2 , for x > 0, so that For the even case, let where E q are the Euler numbers and A (2q) are the Euler (or secant) numbers A000364, given in the online Encyclopedia of integer sequences, [16].
For the odd case, let and using x = tan θ 2 on the second integral, we have The proof of the theorem is finished.
We know consider the second case of the integral (1.1). A. Sofo and Theorem 3 follows.Some examples follow.

Conclusions
We have carried out a systematic study of integrals containing a log tangent function in terms of Euler sums.We believe most of our results are new in the literature and have given many examples some of which are not amenable to a mathematical computer package.

π 2 0fπ
(x) ln (tan x) dx can be approximated by a finite sum involving the Riemann zeta function at odd positive integers.The authors prove that for any polynomial P (x) 2k−1 c k (P ) ζ (2k + 1) ,