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Unformatted text preview: 211 Part C. FOURIER ANALYSIS. PARTIAL DIFFERENTIAL EQUATIONS (PDEs) CHAPTER 11 Fourier Series, Integrals, and Transforms Change The first two sections are now combined into a single section, giving a better and somewhat faster start, with more emphasis on the essential ideas and facts. SECTION 11.1. Fourier Series, page 478 Purpose. To derive the Euler formulas (6) for the coefficients of a Fourier series (5) of a given function of period 2 p , using as the key property the orthogonality of the trigonometric system. Main Content, Important Concepts Periodic function Trigonometric system, its orthogonality (Theorem 1) Fourier series (5) with Fourier coefficients (6) Representation by a Fourier series (Theorem 2) Comment on Notation If we write a /2 instead of a in (1), we must do the same in (6a) and see that (6a) then becomes (6b) with n 5 0. This is merely a small notational convenience (but may be a source of confusion to poorer students). Comment on Fourier Series Whereas their theory is quite involved, practical applications are simple, once the student has become used to evaluating integrals in (6) that depend on n . Figure 257 should help students understand why and how a series of continuous terms can have a discontinuous sum. Comment on the History of Fourier Series Fourier series were already used in special problems by Daniel Bernoulli (1700–1782) in 1748 (vibrating string, Sec. 12.3) and Euler (Sec. 2.5) in 1754. Fourier’s book of 1822 became the source of many mathematical methods in classical mathematical physics. Furthermore, the surprising fact that Fourier series, whose terms are continuous functions, may represent discontinuous functions led to a reflection on, and generalization of, the concept of a function in general. Hence the book is a landmark in both pure and applied mathematics. [That surprising fact also led to a controversy between Euler and D. Bernoulli over the question of whether the two types of solution of the vibrating string problem (Secs. 12.3 and 12.4) are identical; for details, see E. T. Bell, The Development of Mathematics, New York: McGraw-Hill, 1940, p. 482.] A mathematical theory of Fourier series was started by Peter Gustav Lejeune Dirichlet (1805–1859) of Berlin in 1829. The concept of the Riemann integral also resulted from work on Fourier series. Later on, these series became the model case in the theory of orthogonal functions (Sec. 5.7). An English translation of Fourier’s book was published by Dover Publications in 1955. im11.qxd 9/21/05 12:33 PM Page 211 212 Instructor’s Manual SOLUTIONS TO PROBLEM SET 11.1, page 485 2. 2 p , 2 p , p , p , 2, 2, 1, 1 4. There is no smallest p . 0. 6. ƒ( x 1 p ) 5 ƒ( x ) implies ƒ( ax 1 p ) 5 ƒ( a [ x 1 ( p / a )]) 5 ƒ( ax ) or g [ x 1 ( p / a )] 5 g ( x ), where g ( x ) 5 ƒ( ax )....
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