A Two-Parameter Trigonometric Series

Transcription

1 A Two-Parameter Trigonometric Series Xiang-Qian Chang Massachusetts Coege of Pharmacy and Heath Sciences Boston MA 25 Let us consider the foowing dua questions. A. Givenafunction fx find its Fourier series α 2 + α k cos kx + β k sin kx. B. Given a trigonometric series A k cos kx + B k sin kx find a function f x such that the given trigonometric series is its Fourier series then we can find the sum of the given trigonometric series. Upon earning the theory of Fourier series we know questions of type A are straightforward but questions of type B are not for a given trigonometric series may not even be a Fourier series as k=2 sin kx n k shows et aone to find a such function. Nevertheess our eriences in doing many type A exercises are cruciay hepfu when tacking type B probems. For instance if you are asked to find the sum of the trigonometric series coskx x [π] your eriences of cacuating the Fourier series of f x = x 2 on [ π π] which gives you x 2 = π k cos kx and of f x = x on [π] which gives you x = π 2 4 π cos2k x 2k 2 wi hep you to figure out that for x [π] coskx = 2 3x 2 6π x + 2π c THE MATHEMATICAL ASSOCIATION OF AMERICA

2 However in most such probems there is ony one parameter invoved. In this short note we are going to obtain the sum of a two-parameter trigonometric series namey the sum of cos 2kπ 2kπ cos n for integer parameters and n by carefuy ocating a function f x and cacuating its Fourier series. How did we find this function? By trying many type A exercises. First et us consider the genera term a n for the periodic sequence }{{}}{{} If = 2 we know a n = 2 n + n = 2...For the genera case we show a n = 2πi n k. 3 In fact it is a direct coroary of the foowing simpe emma. Lemma. 2πi n { if n; k = otherwise. Proof. If n2πi n k = and we get 2πi n k =. If n2πi n k = and we have 2πi n k = 2πi n k 2πi n = 2πi n =. Next et us consider a n from a different ange. Define a continuous function f x on [ ] as f x = { x + for x ; for x ; x + for x. It is easy to see that f x is the inear interpoation of the sequence in 2. Now extend this f x first eveny to [ ] then periodicay to the whoe rea ine R. Denote the extended function as f x. The famous Dirichet-Jordan theorem see [] assures the convergence of its Fourier series to f x; hence a n = f n. VOL. 36 NO. 5 NOVEMBER 25 THE COLLEGE MATHEMATICS JOURNAL 49

3 Now et us cacuate the Fourier series of the even function f x. By the Euer- Fourier formuas we have α = 2 = 2 f x dx = = 2 2 x dx + x + dx and α m = 2 = 2 f mπ x cos x dx { mπ mπ } x + cos x dx + x + cos x dx. 4 It is easy to check that mπ cos x dx = mπ mπ sin and mπ cos x = mπ m+2 sin mπ and by integrating by parts we aso have the foowing: mπ x cos mπ x cos x dx = x dx = sin mπ { m+ sin mπ mπ Now putting a these into 4 we get that Therefore mπ + mπ mπ cos mπ + mπ mπ m cos for m odd; α m = 2kπ cos for m = 2k. π 2 f x = α 2 + m= = + π 2 mπ α m cos x cos 2kπ 2kπ cos x. }. 4 c THE MATHEMATICAL ASSOCIATION OF AMERICA

4 Thus a n = f n = + π 2 cos 2kπ 2kπ cos n. Comparing the above with 3 we have the foowing resut. Theorem. cos or equivaenty cos 2kπ 2kπ 2kπ cos n = π 2 2 2πi n k 2kπ π 2 for n; cos n = 2 π 2 otherwise. 2 5 Remarks.. Letting n = and = 2 we get the we-known identity k k = π which in turn impies the other we-known identity k = π Letting n = in the theorem and using the fact we can easiy get 2kπ k cos 2 = π 2 6 = π 2 6 π 2 which in essence is a different version of what we got in. In contrast to there where we need two Fourier ansions here ony one function f x is needed to derive a much more genera resut The emma athough simpe is usefu in cassica number theory. For instance by this very emma the number of soutions of f x...x n Nmod x j can be ressed as... x = x n = 2πi f x...x n N k For this and reated rich discussions in cassica number theory see [2].. VOL. 36 NO. 5 NOVEMBER 25 THE COLLEGE MATHEMATICS JOURNAL 4

5 Acknowedgments. Finay the author wishes to ress his sincere thanks to the referee and the editors for their suggestions. References. G. Kambauer Mathematica Anaysis Marce Dekker L. G. Hua Introduction to Number Theory Springer-Verag 982. Coming in the December issue of Mathematics Magazine ARTICLES The Lost Cacuus : Tangency and Optimization without Limits Jeff Suzuki Basketba Beta and Bayes Matthew Richey and Pau Zorn The Least-Squares Property of the Lanczos Derivative Nathania Burch Pau E. Fishback and Russe A. Gordon NOTES Honey Where Shoud We Sit? John A. Frohiger Proof Without Words: Aternating Sums of Odd Numbers Arthur T. Benjamin A Short Proof of Chebychev s Upper Bound Kimbery Robertson and Wiiam Staton Recounting the Odds of an Even Derangement Arthur T. Benjamin Curtis D. Bennett and Forence Newberger Voumes of Generaized Unit Bas Xianfu Wang Proof Without Words: A Trianguar Sum Roger B. Nesen Partitions into Consecutive Parts M. D. Hirschhorn and P. M. Hirschhorn Means Generated by an Integra Hongwei Chen Nonattacking Queens on a Triange Eya Lev and Gabrie Nivasch 42 c THE MATHEMATICAL ASSOCIATION OF AMERICA