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## anonymous 3 years ago find the fourier series of this

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1. anonymous

of this

2. AccessDenied

Is there any general form for a Fourier series of a function f(x)? I'm not particularly well-versed in the subject, although I'll try to help... :)

3. AccessDenied

My first observation of the problem is that f(x) goes through three cycles over the interval (-3pi, 3pi)...

4. anonymous

yeah i know

5. anonymous

the first thing I have to find is the fourier coefficient. I think it is 1/4?

6. anonymous

$1/(2\pi)(\int\limits_{-\pi}^{0}0 dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

7. anonymous

which is 1/4

8. AccessDenied

Yep, that appears correct to me.

9. anonymous

$a _{n}=1/\pi \int\limits_{-\pi}^{0}0 dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

10. anonymous

I am not sure if I am doing it right on the last one

11. AccessDenied

The resource I am checking indicate that the formula for a_n would be 1/pi * integral from -pi to pi of f(x) cos(nx) dx

12. AccessDenied

So... $$\displaystyle a_n = \frac{1}{\pi} \left( \int_{-\pi}^{0} 0 \; \textrm{d}x + \int_{0}^{\pi} \frac{1}{\pi} x \cos nx \; \textrm{d}x \right)$$

13. anonymous

yeah so then it is correct

14. anonymous

so then it is 1/pi?

15. anonymous

sin (1/pi)

16. AccessDenied

So I am finding: $$\displaystyle a_n = \frac{\pi n \sin n \pi + \cos \pi n - 1}{\pi^2 n^2}$$

17. anonymous

and then I must do a $b _{n}$

18. anonymous

$b _{n}=1/\pi(\int\limits_{-\pi}^{0}0dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

19. AccessDenied

My resource shows an additional sin(nx) there $$\displaystyle \frac{1}{\pi} \int_{0}^{\pi} \frac{1}{\pi} x \sin nx \; \textrm{d}x$$

20. anonymous

oh it's the same I just forgot to write the nx

21. anonymous

and that -cos (pi/pi)

22. AccessDenied

Sorry, I gotta go for school. I take the calculation of this integral to wolfram... Wolfram evaluates it out as: $$\displaystyle b_n = \frac{\sin \pi n - \pi n \cos \pi n}{\pi^2 n^2}$$ http://www.wolframalpha.com/input/?i=integral+from+0+to+pi+of+1%2Fpi%5E2+x+sin%28n+x%29+dx

23. anonymous

okey. thank you

24. AccessDenied

You're welcome! :)

25. AccessDenied

and good luck, I think you're on the right track. :) I was using this as my resource: http://mathworld.wolfram.com/GeneralizedFourierSeries.html Just the end bit with the formula.

26. anonymous

thank you :)

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