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Evaluation of Some Non-trivial Integrals from Finite Products and Sums

F. M. S. Lima

Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia DF, Brazil

Abstract: In this note, by manipulating the sums obtained from certain finite products of trigonometric functions at rational multiples of π, I put them in the form of Riemann sums. By taking the limit as the number of (equally-spaced) subintervals tends to infinity, I have found exact closed-form results for some non-trivial integrals, e.g. and I also show how the method applies for the prompt evaluation of more complex integrals, such as and Since this approach does not involve any search for primitives, it can be a good alternative to more complex integration techniques.

Keywords: Products of sines, Rational multiples of , Integration techniques

Citation: F. M. S. Lima, Evaluation of Some Non-trivial Integrals from Finite Products and Sums. Turkish Journal of Analysis and Number Theory, 4 (6). 172-176. January 2016. doi:10.12691/tjant-4-6-5

1. Introduction

The continuity (even differentiability) of the functions , , and , except at some isolated points, suggests that the evaluation of the integrals and via the Fundamental Theorem of Calculus should be a straightforward task. However, all these integrals are improper as their integrands have at least one infinite discontinuity in the respective integration interval, and the corresponding indefinite integrals cannot be solved in a (finite) closed-form in terms of elementary functions only. Their exact evaluation usually demands advanced integration techniques, such as the series expansion of the integrand followed by a term-by-term integration (see Sec. 11.9 of Ref. [8]), or the evaluation of a suitable contour integral on the complex plane (associated to the Cauchy's residue theorem), as seen, e.g., in Secs. 4.1 and 4.2 of Ref. [1]. However, these methods present some disadvantages when applied to integrals of `log-trig' functions because the expansion of the integrand in a Taylor series is not possible for and . For , though a series expansion exists, it is not easy to find a closed-form expression for the general term, which makes it difficult to recognize the number it represents. Indeed, the evaluation of a contour integral on the complex plane has the inconveniences of requiring the choice of a suitable path of integration, usually a difficult task, and results in logarithms of complex numbers and non-elementary functions (e.g., dilogarithm, elliptic, and hypergeometric functions), which often makes the final result obscure. This is just what one gets with the use of mathematical softwares, e.g. Maple (release 2015) and Mathematica (release 10), which return the following stodgy result for

where is the dilogarithm function.

In this work, the integrals and will be promptly evaluated from the logarithm of certain products of trigonometric functions at rational multiples of which yields finite sums that can be written in the form of Riemann sums, whose limit as the number of terms tends to infinity will result in closed-form expressions, without any search for primitives. Interestingly, it will be shown that this technique also works for more complex integrals, such as and where is the classical Euler's gamma function, is the digamma function, and is the Clausen integral [5].

2. Some Products of Trigonometric Functions

In Appendix A.3 of Ref. [7], on presenting an elementary proof for the Euler result those authors prove an identity involving being a rational multiple of Let us take their proof as our starting point.

Lemma 1 (Product of tangents). For any integer

Proof. The proof in Ref. [7] follows from a comparison of the De Moivre's theorem and the binomial theorem for For completeness, let us present the main steps. Let be a polynomial equation, with

where Since then which implies that the roots of are just the numbers

The well-known rule for the product of roots of a polynomial equation then yields

The inverse of both sides reads from which the desired result promptly follows.

There in Appendix A.3 of Ref. [7] one also finds the following identity.

Lemma 2 (Product of sines). For any integer N > 1,

Proof. The roots of the polynomial equation are the N-th roots of unity with The Fundamental Theorem of Algebra yields

valid for all Since a division by gives

The limit as yields

[picture] | (1)

From Euler's formula for complex exponentials, one has

Since then

because for all The substitution of this result in Eq. (1) completes the proof.

Other similar products of trigonometric functions may be easily derived.

Lemma 3 (Product of squares of sines and cosines). For any integer N > 1,

and

Proof. From the symmetry relation it follows from Lemma 2 that

valid for all N > 1, from which the product of squares of sines promptly follows. The product of squares of cosines follows from the substitution in the trigono-metric identity

3. Evaluation of Definite Integrals

The general idea underlying my method is to rewrite a known finite product or sum in the form of a Riemann sum with equally-spaced subintervals, whose limit as the number of terms tends to infinity is a definite integral. We begin by applying this procedure to the products of trigonometric functions established in the previous section.

Theorem 1 (Some `log-trig' integrals). The following exact closed-form results hold:

[picture] | (2)

[picture] | (3)

and

[picture] | (4)

Proof. By taking the logarithm of each side of the product found in Lemma 2, one has

[picture] | (5)

This implies that

[picture] | (6)

This can be rewritten as

[picture] | (7)

where and Clearly, the sum at the left-hand side has the form of a Riemann sum in which the grid points are equally spaced by By taking the limit as on both sides and noting that as follows from l'Hopital's rule, one finds

[picture] | (8)

which means that

[picture] | (9)

The change of variable leads to the first integral.{1}

When the above procedure is applied to the product of square of sines in Lemma 3, one finds

The limit as leads to

which is equivalent to the second integral. Similarly, the product of squares of cosines in Lemma 3 yields

The limit as yields which is equivalent to the other (cosine) integral.

Finally, the product of tangents in Lemma 1 leads to

By substituting 2N + 1 = M (hence M > 1 is an odd integer) and then dividing both sides by M, one has

where and The limit as yields

which implies that

Theorem 2 (A less obvious `log-trig' integral). The exact closed-form result

holds for all real values of .

Proof. This closed-form evaluation follows from the finite product

[picture] | (10)

valid for all positive integer N and all This trigonometric identity is proposed as an exercise in Appendix A.3 of Ref. [7] (see its Ex. 6). For all such that sin the logarithmic version of this product reads

[picture] | (11)

A division by N yields

[picture] | (12)

The limit as leads to the desired result.

The above procedure also works for evaluating more complex integrals.

Theorem 3 (A `log-gamma' integral). The exact closed-form result

holds.{²}

Proof. By adding to both sides of Eq. (5), one finds

which promptly simplifies to

[picture] | (13)

where On taking into account the reflection property of the gamma function, i.e. valid for all and dividing both sides by one finds

The limit as then yields

[picture] | (14)

Since this integral expands to the substitution in the latter integral reduces Eq. (14) to

Theorem 4 (Some `impossible' integrals). The exact closed-form results

[picture] | (15)

and

[picture] | (16)

hold.

Proof. From Eq. (6.4) of Ref. [4], a recent paper by Connon, namely

[picture] | (17)

one easily shows the first integral result by reducing it to a Riemann sum and then proceeding as in the above proofs. Similarly, from Eq. (6.14) of Ref. [4], namely

[picture] | (18)

the second integral readily follows.

The integral in Eq. (16), above, can be further manipulated in order to generate two other interesting integrals.

Theorem 5 (Further log-gamma integrals). The exact closed-form results

and

hold.

Proof. On applying the double-angle formula to Eq. (16), it follows that

[picture] | (19)

This implies that

[picture] | (20)

From Theorem 3, the first integral follows. The second integral follows from the identity

Theorem 6 (An integral involving the Clausen function). The exact closed-form result

holds for all integers k > 0.

Proof. In Ref. [6], Nakamura gives a simple proof of{³}

[picture] | (21)

where is the Hurwitz-zeta function and is an integer. On dividing both sides by and taking the limit as one finds

[picture] | (22)

Simple applications of the l'Hopital's rule, together with the substitution reduce Eq. (22) to

[picture] | (23)

from which the desired result promptly follows.

Interestingly, the definite integral in Theorem 6 has the typical form of Fourier coefficients, {4} which has led me to investigate the corresponding cosine integrals, i.e.

[picture] | (24)

which were not explored in literature. After some numerical investigations for small values of k, I have arrived at some conjectures, namely

[picture] | (25)

[picture] | (26)

[picture] | (27)

[picture] | (28)

[picture] | (29)

[picture] | (30)

all accurate to a thousand of decimal places. Apparently, for odd values of k one has p and q being positive integers, whereas for even values of k one has negative rational values. The determination of a general pattern seems to depend on the closed-form evaluation of Eq. (24), which seems to be an open problem. Of course, a proof for any of the above conjectures would be valuable for this line of research.

Notes

1. This result can also be found by applying some trick substitutions, as done in Sec. 12.5 of Ref. [2].

2. This nice result is stamped on the cover of the book Irresistible integrals [2].

3. Typo corrected in the denominator of the first term of his Eq. (2).

4. The Clausen function has a well-known trigonometric expansion, i.e. found by Clausen himself [3].

References

[1]   | M. J. Ablowitz and A. S. Fokas, Complex Variables (2nd ed.), Cambridge University Press, NewYork, 2003.
[2]   | G. Boros and V. H. Moll, Irresistible Integrals, Cambridge University Press, New York, 2004.
[3]   | T. Clausen, Über die Function sinφ+(1/2²)sin2φ+(1/3²)sin3φ+etc., J. Reine Angewandte Mathematik 8, 298-300 (1832).
[4]   | D. F. Connon, Determination of the Stieltjes constants at rational arguments. Available at arXiv: 1505.06516v2 (2015).
[5]   | L. Lewin, Structural properties of polylogarithms, American Mathematical Society, Providence, 1991.
[6]   | T. Nakamura, Some formulas related to Hurwitz-Lerch zeta functions, Ramanujan J. 21, 285-302 (2010).
[7]   | I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers (5th ed.), Wiley, New York, 1991.
[8]   | J. Stewart, Calculus (7th ed.), Brooks/Cole, Belmont (USA), 2012.