Chapter 3 - III Fourier Series Contents 1 Pre-knowledge 1 2...

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Unformatted text preview: III Fourier Series Contents 1 Pre-knowledge 1 2 (Full) Fourier Series 2 3 Fourier Cosine and Sine Series on Interval [0 , L ] 4 4 (Full) Fourier Series on Interval [ a, b ] 5 5 Extension of Functions 5 6 Periodic Solution of non-homogeneous equations 8 1 Pre-knowledge 1. cos( a ± b ) = cos a cos b ∓ sin a sin b. ⇒ sin a sin b = cos( a + b ) − cos( a − b ) 2 , cos a cos b = cos( a + b ) + cos( a − b ) 2 . 2. If f ( x ) is odd on [ − a, a ] (i.e. f ( − x ) = − f ( x )), then ∫ a − a f ( x ) dx = 0; 3. If f ( x ) is even on [ − a, a ] (i.e. f ( − x ) = f ( x )), then ∫ a − a f ( x ) dx = 2 ∫ a f ( x ) dx. 4. If m , n are non-negative integers, and m ̸ = n , then ∫ π − π cos( nx ) cos( mx ) dx = 0 , ∫ π − π sin( nx ) sin( mx ) dx = 0 . Also, even if we drop the restriction m ̸ = n , ∫ π − π cos( nx ) sin( mx ) dx = 0 . 5. If n is a positive integer, then ∫ π − π cos 2 ( nx ) dx = ∫ π − π sin 2 ( nx ) dx = π. 1 2 (Full) Fourier Series Definition . Given a function f ( x ) on the interval [ − L, L ] with f and f ′ piecewise contin- uous, suppose f ( x ) = a 2 + ∞ ∑ n =1 ( a n cos( nπx L ) + b n sin( nπx L )) (1) at every x ∈ [ − L, L ] where f is continuous. This series is called the (full) Fourier series for f ( x ). The coefficients a n ( n ≥ 0) are called the Fourier cosine coefficients, and the coefficients b n ( n ≥ 1) are called the Fourier sine coefficients. Theorem . The Fourier coefficients can be calculated as follows: a n = 1 L ∫ L − L f ( x ) cos( nπx L ) dx n = 0 , 1 , 2 , . . . . (2) b n = 1 L ∫ L − L f ( x ) sin( nπx L ) dx n = 1 , 2 , 3 , . . . . (3) Proof. The coefficient a is the simplest to find: integrating (6) from − L to L , ∫ L − L f ( x ) dx = ∫ L − L a 2 dx + ∞ ∑ n =1 { a n ∫ L − L cos( nπx L ) dx + a n ∫ L − L sin( nπx L ) dx } = ∫ L − L a 2 dx The series on the right vanishes, and we find that a = 1 L ∫ L − L f ( x ) dx. We do the same thing to compute, say, b m , except that first we multiply (6) through by sin( mπx L ). We get ∫ L − L f ( x ) sin( mπx L ) dx = ∫ L − L a 2 sin( mπx L ) dx + ∞ ∑ n =1 a n ∫ L − L cos( nπx L ) sin( mπx L ) dx + b n ∫ L − L sin( nπx L ) sin( mπx L ) dx. What is important to notice is that all of the integrals on the right side vanish, except for the one multiplying b m . The equation for b m becomes b m = 1 L ∫ L − L f ( x ) sin( mπx L ) dx m = 1 , 2 , 3 , . . . ....
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This note was uploaded on 07/11/2011 for the course ECE 45 taught by Professor Lee during the Spring '08 term at Alfred University.

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Chapter 3 - III Fourier Series Contents 1 Pre-knowledge 1 2...

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