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Fourier Series and the Fourier Integral

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Fourier Series and the Fourier Integral CORE Metadata, citation and similar papers at core.ac.uk Provided by K-State Research Exchange FOURIER SERIES AND THE FOURIER INTEGRAL by BILLY GENE CLAYBROOK B. S., Missouri State Teachers College, 1964 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics KANSAS STATE UNIVERSITY Manhattan, Kansas 1966 Approved by: x^ C77\U^^vy,.a.^ /^C^C^^OC-i__^ 20 iff ii Lifill. TABLE OF COOTENTS CX INTRODUCTION 1 FOURIER SERIES OF A FUNCTION 3 CONVERGENCE OF FOURIER SERIES 10 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES 15 '1 THE FOURIER INTEGRAL 20 APPLICATION OF FOURIER SERIES AND INTEGRALS 30 BIBLIOGRAPHY 42 ACKNOWLEDGMENT 43 INTRODUCTION Attention was drawn to trigonometric series in the middle of the eighteenth century when various mathematicians were studying the vi- brations of stretched strings. The mathematical theory of such vi- brations comes down to the problem of solving the partial differential equation o " '^^^ " = b^ ° • general form of a trigonometric series, involving both sines and cosines, arises naturally in con- nection with another physical problem, that of conduction of heat. It was in the course of his study of this problem that J. B. Fourier announced in 1807 that an arbitrary function f (x) could be represented in the form f(x)sr_fo_ ^ y (a^j cos nx ^b^ sin nx) with a^ and b^ given by the integrals ^n»n== -^'^ yUW cos nx dx, n«:0,l,2, ... bn'n = 'n U(^\f(x) sin nx dx, n =1,2 n The trigonometric series given above is called the Fourier series corresponding to f(x) and the constants a^ and b^ are called the Fourier coefficients of the function. We should point out, however, some trigonometric series are not Fourier series. A trigonometric series of the form — -^ \ (flln cos nx -»- b^ sin nx) 2 is not called a Fourier series unless there is some function f (x) from which the coefficients in the series may be calculated by the formulas given above for a^ and b^. The integrals defining a^ and b^ must be meaningful, i.e., the integrals exist either as proper Rieraann integrals, Lebesgue integrals or as improper integrals. Throughout this Report when the Fourier series of a function is being studied the function will be assumed Riemann integrable. Most of the functions arising in ordinary ap- plications are bounded and are continuous except possibly at a finite number of points and are therefore integrable in the sense of Riemann. Fourier series are useful in dealing with periodic functions but if the function is not periodic, it cannot be represented by a Fourier series. There is, however, an expression involving improper integrals instead of infinite series, which under suitable conditions gives a valid representation of a nonperiodic function. Then we say the function is represented by its Fourier integral. The purpose of this Report is to acquaint the reader with some of the most important aspects of the theory and applications of Fourier series and Fourier integrals. Fourier Series of a Function on the Definition 1.1. If f (x) is a function which is defined interval aixib and such that the integrals n dx, n « 0,1,2, ... (1) ajj = ^ (f(x) cos nx f(x) sin nx dx, n S'1,2, ... have a meaning, we form the series ^o + ) (an cos nx •»• b^ sin nx) and call this the Fourier series corresponding to f(x). The function determines its Fourier series quite apart from any consideration as to whether the series converges or, if it does con- verge, whether or not its sum is f(x). If the Fourier series of a con- function f (x) is formed without deciding in advance whether it verges to f(x), then we write c ^° f (x) rsJ -^ \ (a^ cos nx « b^ sin nx) . This notation means merely that the Fourier series on the right cor- responds to the function f(x). The sign r>* can be replaced by the sign = only if we succeed in proving that the series converges and that its sum equals f(x). Before we find the Fourier series corresponding to a particular function, we should give an explanation of how we arrived at the in- tegrals for the Fourier coefficients an and bn given by (1) and (2). First consider the orthogonality of sine and cosine functions. For . any Integer n^O r" (3) \ cos nx dx - (4) 1 sin nx dx = (5) \cos2 nx dx ss 77 -71 Tf (6) sin^ nx dx si . Let m and n denote positive integers. Then 1 (7) Icos nx cos mx dx = (8) sin nx sin mx dx ^ (9) IsinIs in nx cos mx dx =fO for any m and n. Suppose the function f(x) of period 2 TT has the expansion (10) f(x) sr_fo_ . \ (^a^ cos kx +h^ sin kx) It will be assumed that the series (10) and the series to be written later can be integrated term by term. Integrating (10) from -n to H, we obtain , \ bj^ I f (x) dx = j dx + f la^ cos kx dx + sin kxdxj. J By (3) and (4) all the integrals in the sinn vanish, so that f(x) dx = TTaj,. f Next, multiply both sides of (10) by cox nx and integrate the result from -w to n, as before, obtaining (11) I f(x) COS nx dx s ^o I cos nx dx + / [ak I cos kx cos nxdx r ^ I sin kx cox nx dx I By (3) the first integral on the right vanishes. By (7) and (9) all the integrals in the sum vanish except one. Thus (11) reduces to r r I f (x) COS nx dx s a^^ I cos( nx dx. Now by (5) (12) fC ^°^ "* ^*» " -0.1.2, . an»n sa JL__ I ^W •IT Similarly, n (13) bjj - J^ f(x) sin nx dx, nsl,2, ... I Before computing the Fourier series corresponding to a particular function f(x), there is a consideration which will sometimes simplify the calculation of the Fourier coefficients. Definition 1.2. The function f(x) is said to be an even function if f(x)=:f(-x) for every x. This definition implies that the graph of any even function is symmetric with respect to the y-axis. It follows from the interpretation of the integral as an area that for even functions defined and integrable on the interval f"n»n7 t (14) lf(x) dx = 2 f(f(x) dx. Definition 1.3. The function f(x) is said to be odd if f(-x)s:-f(x) for every x. The graph of any odd function f(x) is symmetric with respect to the origin. Thus, for odd functions defined and integrable on the interval f-w, nj. •n (15) flf(x) dx ssO. The following properties are consequences of the definition of even and odd functions: (i) The product of two even or two odd function is an even function, (ii) The product of an even and an odd function is an odd function. , Thus if (j^fjOand Y(x) are even functions, then for f(x)= ^(x) 'ij' M we have f(-x)= ^(-x) V(-x)=0(x)VW = fW. while if ^(x) and ^(x) are odd, f (-X) = ^(-x) V (^-^) = [-<twl [- V(x)J = ^(^> V W = f W' This proves Property (i). On the other hand, if j^(x) is even and Y (x) is odd, then f(-x) = ^ (-X) -^ (-X) - 0(x) f-V^'^)/ * " ^^''^ '^^''^ " "^^''^ which proves Property (ii) Let f(x) be an even function defined on the interval [-H.TlJ . Since cos nx (n»0, 1, 2,...) is an even function, then by Property (i) the function f(x) cos nx is also even. On the other hand, the function sin nx (nsl, 2, ...) is odd, so that the function f(x) sin nx is also odd, by Property (ii) . Then, using (12), (13), (14) and (15), we find that the Fourier coefficients of the even function f(x) are 2 It a^ - .. I f(x) cos nx dx - £(x)f(x) cos nx dx, (16) In =—s I ;:: _2_ n= 0,1,2 ... ^ J , ^ J n (17) bjj -. f (x) sin nx dx =0 , n = 1,2, ... _J_ I •n Therefore, the Fourier series of an even function contains only cosines, i.e., f (x) ru fo_ + \ an cos nx. , 8 The series above is known as the Fourier cosine series. odd function defined on the interval • Now let f (x) be an [-11, IT J Since cos nx (n=0, 1, 2, ...) is an even function, then by Property (ii) f(x) cos nx is an odd function. On the other hand, the function sin nx (n=l,2, ...) is odd and by Property (i) the function f(x) sin Y»X K is even. Then, using (12), (13), (14) and (15) we find that the Fourier coefficients of the odd function f(x) are (x) cos nxdxsO, n=0, 1,2, ... J-n (x) sin nx dx s 2 f(x) sin nx dx, n = l,2, .. '— n Therefore, the Fourier series of an odd function contains only sines, i.e. f (x) ro L-) bn sin nx. This series becomes the Fourier since series. The following example illustrates the procedure for finding the Fourier series corresponding to a particular function. Example ; Find the Fourier series corresponding to the function f (x) s:x^/4 when -nix-iTl. The function is even, so b^ — and an x^ cos nx dx.
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