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Homework2 - Physics 401 Homework#2 1 Two forms of Fourier...

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Physics 401 - Homework #2 1) Two forms of Fourier Series (three points). If f(x) is periodic, with period 2L, and square integrable between (-L,L), then we can represent it as a linear combination of the functions L x in / exp( (a Fourier Series): L x in n n n e c x f / ) (  for some set of coefficients {c n }. If f(x) is real, then we can also represent the function this way: 1 0 sin cos 2 ) ( n n n L x n b L x n a a x f where the relationship between the two forms is: 0 0 ) ( ) ( 2 ) ( c a c c i b c c a n n n n n n Show that these two forms are equivalent by substituting for a n , b n , and a 0 in the second form and recovering the first form. Hint: use Euler's formula to convert sines and cosines into exponentials. 2) Fourier Series of a square wave. We can write a square wave so it's an odd function: L x x L x f 0 , 1 0 , 1 ) ( , periodic with period 2L.
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