Recall that a piecewise continuous function has only

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Unformatted text preview: a jump discontinuity, the one-sided limits exist and we use the notation fa lim f x fa xla lim f x xla 8 Fourier Convergence Theorem If f is a periodic function with period 2 and f and f are piecewise continuous on , , then the Fourier series (7) is convergent. The sum of the Fourier series is equal to f x at all numbers x where f is continuous. At the numbers x where f is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is 1 2 fx fx If we apply the Fourier Convergence Theorem to the square-wave function f in Example 1, we get what we guessed from the graphs. Observe that f0 lim f x 1 xl0 and f0 lim f x 0 xl0 and similarly for the other points at which f is discontinuous. The average of these left and right limits is 1 , so for any integer n the Fourier Convergence Theorem says that 2 1 2 2 k1 2k 1 sin 2k (Of course, this equation is obvious for x 1x n .) fx 1 2 if n if x n n 6 ■ FOURIER SERIES Functions with Period 2L If a function f has period other than 2 , we can find its Fourier series by making a change f x for all x. If we let of variable. Suppose f x has period 2 L, that is f x 2 L t x L and tt fx f Lt then, as you can verify, t has period 2 and x series of t is a0 L corresponds to t a n cos nt . The Fourier bn sin nt n1 where 1 2 a0 1 an y y t t dt t t cos nt dt If we now use the Substitution Rule with x we have the following 9 Lt nx L a n cos n1 y t t sin nt dt , then t If f is a piecewise continuous function on a0 1 bn x L, dt L d x, and L, L , its Fourier series is bn sin nx L where Notice that when L these formulas are the same as those in (7). ■■ 1 2L a0 and, for n an 1 L y L f x dx L 1, y L L f x cos nx L dx 1 L bn y L L f x sin nx L dx Of course, the Fourier Convergence Theorem (8) is also valid for functions with period 2 L. EXAMPLE 2 Find the Fourier series of the triangular wave function defined by f x x for 1 x 1 and f x 2 f x for all x. (The graph of f is shown in Figure 3.) For which values of x is f x equa...
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This note was uploaded on 12/05/2011 for the course MATH 41 taught by Professor Bray,c during the Spring '08 term at Duke.

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