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M257-316Notes_Lecture13

M257-316Notes_Lecture13 - Chapter 9 Lecture 13 Fourier...

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Chapter 9 Lecture 13 - Fourier Series We consider the expansion of the function f ( x ) of the form f ( x ) a 0 2 + n =1 a n cos nπx L + b n sin nπx L = S ( x ) (9.1) where a n = 1 L L L f ( x ) cos nπx L dx a 0 2 = 1 2 L L L f ( x ) dx = average value of f . b n = 1 L L L f ( x ) sin nπx L dx (9.2) Note: 1. Note that cos L ( x + λ ) = cos nπx L provided nπλ L = 2 π , λ = 2 L n and similarly sin L ( x + 2 L ) = sin nπx L . Thus each of the terms of the Fourier Series S ( x ) on the RHS of (10.1) is a periodic function having a period 2 L . As a result the function S ( x ) is also periodic. How does this relate to f ( x ) which may not be periodic? The function S ( x ) represented by the series is known as the periodic extension of f on [ L, L ]. 61
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Lecture 13 - Fourier Series 2. If f (or its periodic extension) is discontinuous at a point x 0 then S ( x ) converges to the average value of f across the discontinuity. S ( x 0 ) = 1 2 f ( x + 0 ) + f ( x 0 ) (9.3) Example 9.1 f ( x ) = 0 π < x < 0 L = π x 0 x π (9.4) 62
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a 0 = 1 π π π f ( x ) dx = 1 π π 0 x dx = π 2 (9.5) a n = 1 π π π f ( x ) cos( nx ) dx = 1 π π 0
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