# Use Fourier series to show

1. Oct 10, 2011

### zheng89120

1. The problem statement, all variables and given/known data

Consider the function:

f(x) = {0 if 0<x<L/2
x-L/2 if L/2<x<L}

Define a periodic extension, obtain the complex Fourier series, and show that Ʃ1/(2m+1)^2 = pi^2/8...

2. Relevant equations

complex Fourier series

3. The attempt at a solution

I defined it as an even function by reflecting the function over the y-axis.

I did some calculations which yielded a complex Fourier series coefficient of:

cn = L[ exp(-i*pi*n)/(-2i*pi*n) + exp(-i*pi*n/2)/(pi2*n2) ]

not sure if this is correct, and how to get the fact that Ʃ1/(2m+1)2 = pi2/8

P.S. Sorry I forgot to add that they wanted: Define a periodic extension over period 2L

Last edited: Oct 10, 2011
2. Oct 10, 2011

### Staff: Mentor

I don't think this is what they had in mind.

By extending the function, I believe they wanted you to repeat the same pattern that's in the interval [0, L].

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