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# Simple fourier series problem Watch

1. It's really confusing me. I thought when the function is even then only the bn terms remain? When I sub in f(-x) here I get f(x) in both cases. I tried therefore to find the b[SUB]n[SUB] term but I got zero:

What are my limits? I thought the usual integral is between -pi and pi and multiplied by 1/pi. Here I split it up into 2 integrals: The first integrand was -x2cos(nx) multiplied by 1/pi and integrated between -pi and 0. The second integrand was x2cos(nx) multiplied by 1/pi and integrated between 0 and pi. Is this wrong?
2. The function is odd.

Therefore the vanish. Moreover so . Thus you have

.

For the limits you can take any interval of length since the function is -periodic. is fine. Therefore

.

Your splitting of the integral is fine...but the functions you're integrating are and (respectively) not cos.

Edit again: still doesn't work...draw your own picture.
3. (Original post by jb444)
The function is odd.

Therefore the vanish. Moreover so . Thus you have

.

For the limits you can take any interval of length since the function is -periodic. is fine. Therefore

.

Your splitting of the integral is fine...but the functions you're integrating are and (respectively) not cos.

Edit again: still doesn't work...draw your own picture.
Thanks, the link worked for me and I understand the rest of your post but I still don't get the graph. It says -x^2 for x less than 0. Surely it should be below the x-axis? also i worked out a0 and got 2/3 pi^2 but i thought it was 0? i followed all the normal rules for calculating a0
4. Yes, it should be below the x axis for $x < 0$ (I deleted the link since it showed the wrong graph, wolfram doesn't load quite the same thing if you copy the URL it seems). It's this extended periodically

To see that you have that

so it'll cancel. Alternatively you can note that and for each , so substituting all this into the Fourier series equation you must have that as well.

Yet another edit: that's still the wrong graph, it should be sgn(x) x^2, obviously.

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Updated: March 31, 2013
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