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Ordinary &amp; Partial Differential Equations - Reynolds (2000) - Chapter 12 - Introduction to Fouri

# Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 12 - Introduction to Fouri

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CHAPTER 12 Introduction to Fourier Series I n the previous chapter, we developed the separation of variables technique to construct the solutions of homogeneous Dirichlet and Neumann problems. In each example, we were able to construct series representations of the solutions provided that the initial conditions themselves had special series representations (i.e., Fourier sine and cosine series). In this chapter, we will study Fourier series in greater depth, addressing three principal questions: ± Which functions φ ( x ) have Fourier series representations? ± Given a function φ ( x ) that does have such a representation, how can we calculate the coef±cients that appear within the series? ± Can we really be sure that these series converge to φ ( x ) ? Before beginning our study of Fourier series, let us brie²y recall another type of series representation that you likely studied in calculus: Taylor series. Suppose that a function f ( x ) has in±nitely many derivatives in some open interval containing the point x = a . In calculus, you learned that the Taylor series for f ( x ) centered at a is given by n = 0 f ( n ) ( a ) n ! ( x - a ) n . Computing the constants f ( n ) ( a ) requires that we ±rst calculate the n th derivative of f and then evaluate at x = a . For example, suppose we wish to compute the Taylor series for f ( x )= e x centered at 0. Since f ( n ) ( x e x for all n , it follows that f ( n ) ( 0 1 for all n . Thus, the Taylor series for e x centered at 0 is given by n = 0 1 n ! x n . 330

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introduction to fourier series 331 The ratio test shows that this series converges for all real x . Moreover, the series really does converge to e x for all real x . Recall that an inFnite series is deFned as a limit of partial sums; e.g., e x = lim N N n = 1 1 n ! x n . Notice that partial sums of a Taylor series are nothing more than polynomials , as are the factors ( x - a ) n that appear in each individual term. In this sense, a Taylor series essentially represents a function f ( x ) as a sum of polynomials. ±ourier series offer another way of representing functions as inFnite series. Un- like Taylor series, which use polynomials as “building blocks”, ±ourier series are sums of sine and cosine functions. More speciFcally, a ±ourier series effectively decomposes a function φ ( x ) into a sum of sine and cosine functions all of which have frequencies that are integer multiples of some “fundamental frequency”. Defnition 12 . 0 . 2 . Let L > 0. A Fourier sine series is a series of the form n = 1 A n sin ± n π x L ² ( 0 < x < L ) .( 12 . 1 ) A Fourier cosine series is a series of the form A 0 2 + n = 1 A n cos ± n π x L ² ( 0 < x < L ) 12 . 2 ) A (full) Fourier series is a series of the form A 0 2 + n = 1 \$ A n cos ± n π x L ² + B n sin ± n π x L ²% ( - L < x < L ) 12 . 3 ) Notice that the interval over which the full ±ourier series is deFned is symmetric about x = 0, whereas the sine and cosine series are deFned for ( 0 < x < L ) .
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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 12 - Introduction to Fouri

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