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 + X n =1 a n cos ³ nπx L ´ + b n sin ³ nπx L ´ = S ( x ) (9.1) where a n = 1 L L Z L f ( x ) cos ³ nπx L ´ dx a 0 2 = 1 2 L L Z L f ( x ) dx = average value of f . b n = 1 L L Z L f ( x ) sin ³ nπx L ´ dx (9.2) Note: 1. Note that cos ³ L ( x + λ ) ´ = cos ³ nπx L ´ provided nπλ L =2 π , λ = 2 L n and similarly sin ³ L ( x +2 L ) ´ =sin ³ 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.
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This note was uploaded on 05/04/2011 for the course MATH 25 taught by Professor Lo during the Spring '11 term at BC.

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

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