c21mudd - ) and cos( ). Therefore 8 cos(3 t ) cos(5 t ) f (...

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± ± 18.03 Muddy Card responses, April 3, 2006 1. A number of people were confused by my derivation of the Fourier coefficients of the function f ( t ), even, periodic, period 2 π , with f ( t ) = 4 for 0 < t < π/ 2 and f ( t ) = 0 for π/ 2 < t < π . I think the process of expressing this in terms of the standard square wave, f ( t ) = 2 + 2sq( t + ( π/ 2)), was pretty clear. Note that the gap in the square wave is 2, while the gap in the graph of f ( t ) is 4: hence the factor of 2. Then you get 8 sin(3( t + ( π/ 2))) f ( t ) = 2 + sin( t + ( π/ 2)) + π 3 + · · · so we have to understand sin( n ( t +( π/ 2))) = sin( nt +( nπ/ 2)). I did not explain that well. Think of the nt as a single unit, so we want to understand sin( θ + ( nπ/ 2)), especially for n odd. When n = 1, this is sin( θ + ( π/ 2)), which is cos( θ )—look at the graphs! When n = 3, this is sin( θ + (3 π/ 2) = cos( θ )—look at the graphs again! When n = 5, sin( θ + (5 π/ 2)) = sin( θ + ( π/ 2)), and we are back in the n = 1 case again. The values thus alternate between cos(
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Unformatted text preview: ) and cos( ). Therefore 8 cos(3 t ) cos(5 t ) f ( t ) = 2 + cos( t ) + . 3 5 2. When I was computing the Fourier coecients for sq( t ), I wanted to evaluate cos( n ) for various values of n . I started with n = 0, despite the fact that b 0 does not occur as a Fourier coecient. This was because I knew that these coecient would repeat after a while, so by starting early I would see the repetition quicker. 3. There were some questions about Fourier series for more general functionsby which I guess you meant non-periodic functions. This is possible, but you have to use sin( t ) for all values of , not just values which are multiplies of some fundamental circular frequency. This is the Fourier transform, and it is closely related to the Laplace transform. 4. Some people wanted to hear more about the Gibbs eect: please see the Supple-mentary Notes, 16....
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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