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Fourier Series_Summary.pdf

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MATH 251

Summary of Fourier Series Suppose f is a piecewise continuous periodic function of period 2L, then f has a Fourier series representation a0 ∞  n x nπx  f x) = +∑  ancos + bnsin  . 2 n=1 L L  Where the coefficients a’s and b’s are given by the Euler-Fourier formulas: 1 L m x am= ∫ f x)cos dx , m = 0, 1, 2, 3, … L −L L L b = 1 f x)sinnπx dx n L −L L , n = 1, 2, 3, … The Fourier Convergence Theorem Theorem: Suppose f and f ′ are piecewise continuous on the interval −L ≤ x ≤ L. Further, suppose that f is defined elsewhere so that it is periodic with period 2L. Then f has a Fourier series as stated above whose coefficients are given by the Euler-Fourier formulas. The Fourier series converge to f (x) at all points where f is continuous, and to   x→c−f (x)+lx→c+ (x)/2   at every point c where f is discontinuous. Fourier Cosine and Sine Series If f is an even periodic function of period 2L, then its Fourier series contains only cosine (include, possibly, the constant term) terms. It will not have any sine term. That is, its Fourier series is of the form a0 ∞ nπx f ( )= + ∑ an cos . 2 n 1 L Its Fourier coefficients are determined by: L 2 mπx am = ∫ f x)cos dx , m = 0, 1, 2, 3, … L 0 L b n= 0, n = 1, 2, 3, … If f is an odd periodic funct
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