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Ebook Advanced engineering mathematics (7th edition): Part 2

(BQ) Part 2 book "Advanced engineering mathematics" has contents: Fourier series, the fourier integral and transforms, special functions and eigenfunction expansions, the wave equation, the heat equation, the potential equation, complex integration, singularities and the residue theorem,.and other contents. | PA R T 4 Fourier Analysis, Special Functions, and Eigenfunction Expansions CHAPTER 13 Fourier Series CHAPTER 14 The Fourier Integral and Transforms CHAPTER 15 Special Functions and Eigenfunction Expansions Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. October 14, 2010 14:57 THM/NEIL Page-425 27410_13_ch13_p425-464 CHAPTER 13 WHY FOURIER SERIES? THE FOURIER SERIES OF A FUNCTION SINE AND C O S I N E S E R I E S I N T E G R AT I O N A N D D I F F E R E N T I AT I O N Fourier Series In 1807, Joseph Fourier submitted a paper to the French Academy of Sciences in competition for a prize offered for the best mathematical treatment of heat conduction. In the course of this work Fourier shocked his contemporaries by asserting that “arbitrary” functions (such as might specify initial temperatures) could be expanded in series of sines and cosines. Consequences of Fourier’s work have had an enormous impact on such diverse areas as engineering, music, medicine, and the analysis of data. 13.1 Why Fourier Series? A Fourier series is a representation of a function as a series of constant multiples of sine and/or cosine functions of different frequencies. To see how such a series might arise, we will look at a problem of the type that concerned Fourier. Consider a thin homogeneous bar of metal of length π , constant density and uniform cross section. Let u(x, t) be the temperature in the bar at time t in the cross section at x. Then (see Section 12.8.2) u satisfies the heat equation ∂ 2u ∂u =k 2 ∂t ∂x for 0 0. Here k is a constant depending on the material of