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Trigonometric Series Via Laplace Transforms

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Trigonometric Series Via Laplace Transforms 376 MATHEMATICS MAGAZINE Trigonometric Series via Laplace Transforms COSTAS J. EFTHIMIOU Department of Physics University of Central Florida Orlando, FL 32816 [email protected] In another Note in this MAGAZINE [2], I presented a method that uses the Laplace transform to find exact values for a large class of convergent series of rational terms. Recently, also in the MAGAZINE, Lesko and Smith [3] revisited the method and demonstrated an extension of the original idea to additional infinite series. My inten- tion in this note is to illustrate the power of the technique in the case of trigonometric series. Trigonometric series Trigonometric series play a vital role in mathematics and physics. Many results are known but most of them can be obtained only via Fourier analysis. For example ∞ cos(nx) x =−ln 2sin , 0 < x < 2π, (1) n=1 n 2 ∞ cos(nx) 3x 2 − 6πx + 2π 2 = , 0 ≤ x ≤ 2π, (2) 2 n=1 n 12 ∞ sin(nx) π − x = , 0 < x < 2π, (3) n=1 n 2 ∞ cos(nx) x (−1)n =−ln 2cos , −π<x <π, (4) n=1 n 2 ∞ cos(nx) 3x 2 − π 2 (−)n = , −π ≤ x ≤ π, (5) 2 n=1 n 12 ∞ cos((2n + 1)x) 1 x =− , < x <π, ( + ) ln tan 0 (6) n=0 2n 1 2 2 ∞ cos((2n + 1)x) π 2 − 2πx = , 0 ≤ x ≤ π. (7) ( + )2 n=0 2n 1 8 Davis [1] offers these and many additional results. Given a function, expanding it in a trigonometric series is relatively straightforward. However, it is almost impossible to guess the function that would generate a given trigonometric series as its Fourier series. For example, given the series ∞ sin((2n + 1)x) π π (−1)n , − ≤ x ≤ , ( + )2 n=0 2n 1 2 2 it is not easy to guess that the function πx f (x) = 4 VOL. 79, NO. 5, DECEMBER 2006 377 gives rise to it. Proving that the guess is correct requires additional work. On the other hand, the method I presented [2], with the extension of Lesko and Smith [3], yields the results in a straightforward manner with no ad hoc guessing. The method As Lesko and Smith pointed out, the original method can be applied to v series of the form n∈I un n where I is a subset of the integers. In series of this form, it is often convenient to write only one of the factors, say vn, as a Laplace transform of a function f (t) +∞ −nt vn = e f (t) dt. 0 Then +∞ −nt unvn = un e f (t) dt. n∈I n∈I 0 Assuming that the order of the operations of summation and integration can be ex- changed +∞ −nt unvn = un e f (t) dt. n∈I 0 n∈I In this note we shall always exchange the order of the two operations, assuming that the reader knows how to justify this step. Details on this may be found in the original papers [2, 3]. ( ) = −nt If one can find an explicit function h t n∈I un e , then this leads to a simple integral representation of the initial series: +∞ unvn = h(t) f (t) dt. n∈I 0 If, furthermore, the integration can be performed, then analytic expressions for the sums of the initial series are obtained. Elementary trigonometric sums To apply the method to trigonometric series, we need to be able to handle series of the form S = sin(nx) e−nt, ∈ n I C = cos(nx) e−nt. n∈I These summations are performed easily using complex notation: C + iS = einx e−nt. n∈I In particular, when I = N∗ ={1, 2,...}, assuming that x is a real number and t > 0, we find 378 MATHEMATICS MAGAZINE ∞ e−t sin x sin(nx) e−nt = , − −t + −2t n=1 1 2cosxe e ∞ e−t (cos x − e−t ) cos(nx) e−nt = . − −t + −2t n=1 1 2cosxe e Similarly we can compute other sums. Also, it is possible to start with these sums and, by changes in the argument x and simple manipulations, derive formulæ for new sums, such as sums over the even or odd integers only. The reader may wish to experiment with this idea. Trigonometric series via the Laplace transform We are now ready to find exact sums for more complicated trigonometric series. We shall demonstrate the method with two examples, namely (1) and (3). The other formulas may be obtained similarly. Alternatively, they may be computed using algebraic and integral operations on (1) and (3). For example, equation (2) may be proved by integrating (3) and using the well ∞ / 2 = π 2/ known sum n=1 1 n 6. The last sum, in turn, may be obtained easily using my original method [2] or various other techniques. The approach described here, however, allows us to derive equation (2) without reference to any other sum, assuming that one can integrate the necessary functions, perhaps using a table of integrals. Since tables of integrals are widely available and they are quite extensive (for instance, [4]), the method seems to be quite effective and straightforward. • We start with the series ∞ cos(nx) ν n=1 n where ν ∈ N∗ and 0 < x < 2π,ifν = 1, or 0 ≤ x ≤ 2π,ifν>1. Using 1 1 ∞ = −nt ν−1 , ν e t dt n (ν − 1)! 0 we write ∞ ∞ cos(nx) 1 ∞ = cos(nx) e−nt t ν−1 dt ν (ν − )! n=1 n 1 n=1 0 ∞ 1 ∞ = (nx)e−nt t ν−1 dt (ν − )! cos 1 0 n=1 1 ∞ e−t (cos x − e−t ) = t ν−1 dt. −t −2t (ν − 1)! 0 1 − 2cosxe + e With the change of variables u = e−t , the integral assumes a more compact form ∞ cos(nx) (−1)ν−1 1 cos x − u = (ln u)ν−1 du. ν (ν − )! − + 2 n=1 n 1 0 1 2cosxu u VOL. 79, NO. 5, DECEMBER 2006 379 For ν = 1 ∞ ( ) 1 ( − + 2) cos nx =−1 d 1 2cosxu u − + 2 n=1 n 2 0 1 2cosxu u 1 1 x =− ln(1 − 2cosxu+ u2) =−ln 2sin . 2 0 2 • Following the same steps for the series ∞ ( ) sin nx , ν n=1 n we can arrive at the integral representation ∞ sin(nx) (−1)ν−1 1 (ln u)ν−1 = sin x du. ν (ν − )! − + 2 n=1 n 1 0 1 2cosxu u In particular, for ν = 1 ∞ sin(nx) 1 1 = sin x du ( − )2 + 2 n=1 n 0 u cos x sin x u − cos x 1 sin x π − x = −1 = −1 = . tan tan − sin x 0 1 cos x 2 Conclusion Although the results presented in this paper are not new, we hope that readers will appreciate the ease and transparency of the method. Given any series such as those described in the original articles of Efthimiou [2], Lesko and Smith [3], and this note, the steps are well-defined and require no special tricks. However, traditional methods do vary from series to series, ingenious tricks may be necessary to be intro- duced, and some (or a lot) guessing might be required. We invite the reader to verify our claim by using the Laplace transform technique to find exact sums for her favorite series (of one of the types under discussion) and then compare with the traditional methods. Acknowledgment. While this article was under review, we received a message from Harvey J. Hindin, who pointed out that Albert D. Wheelon had also used the idea of integral transformations to compute exact values for infinite series [5]. We are grateful to one of the referees who read the article with great care and dedication. Although typesetting had introduced many typographical errors into the document, the penetrating and thorough reading of the referee helped very much in detecting them. REFERENCES 1. H.F. Davis, Fourier series and orthogonal functions, Dover, New York, 1989. 2. C. Efthimiou, Finding exact values for infinite sums, this MAGAZINE 72 (1999), 45–51. 3. J.P. Lesko and W.D. Smith, A Laplace transform technique for evaluating infinite series, this MAGAZINE 76 (2003), 394–398. 4. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition, Academic Press, New York, 1980. 5. A.D. Wheelon, Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day Publishers, San Francisco, 1968..
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