tailieunhanh - Ebook Theory and problems of fourier analysis with applications to boundary value problems: Part 2

(BQ) Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. | chapter 5 Fourier Integrals and Applications THE NEED FOR FOURIER INTEGRALS In Chapter 2 we considered the theory and applications involving the expansion of a function f x of period 2L into a Fourier series. One question which arises quite naturally is what happens in the case where L - 00 We shall find that in such case the Fourier series becomes a Fourier integral. We shall discuss Fourier integrals and their applications in this chapter. THE FOURIER INTEGRAL Let us assume the following conditions on f x 1. f x and f x are piecewise continuous in every finite interval. 2. I dx converges . f x is absolutely integrable in 00 oo . z 00 Then Fourier s integral theorem states that A a cos aX B a sin ax da 0 1 r A a 3 I f x cos aX dx 7T 00 where If . B a - I f x sin ax dx 7T -00 Ơ 2 The result 1 holds if a is a point of continuity of fix . If a is a point of discontinuity .________. f x 0 f x - 0 ________. ______. x . we must replace f x by -------- -------- as in the case of Fourier series. Note that the above conditions are sufficient but not necessary. The similarity of 1 and 2 with corresponding results for Fourier series is apparent. The right-hand side of 1 is sometimes called a Fourier integral expansion of f x . EQUIVALENT FORMS OF FOURIER S INTEGRAL THEOREM Fourier s integral theorem can also be written in the forms f eiax da du ẩ 4 0 eia z- du da 80 CHAP. 5 FOURIER INTEGRALS AND APPLICATIONS 8Í where it is understood that if f x is not continuous at X the left side must be replaced by f x 0 f x-0 These results can be simplified somewhat if f x is either an odd or an even function and we have 2 f - 1 sin aX da z 00 1 f u sin au du 0 if f x is odd 5 f x 2 p 1 cos aX da 7T 7o s 00 1 fịụ cos au du 0 if f x IS even ff FOURIER TRANSFORMS From 4 it follows that if F e u du 1 r then f x I F 0 eiax da úilĩ J 00 7 S The function F a is called the Fourier transform of f x and is sometimes written F a F f x . The function f x is the inverse Fourier transform of F a .