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Mindscrape

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I was going through trying to solve various Fourier problems, and I came across this one.

[tex]f(x) = \left\{\begin{array} {c}0 \ \ \mbox{for} \ \ - \pi <x<0 & x \ \ \mbox{for} \ \ 0<x<\pi[/tex]

Here is how far I have gotten, using that

[tex]a_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xcosnx dx[/tex]

and

[tex]b_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xsinnx dx[/tex]

arriving at

[tex] f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + (unknown) [/tex]

So, I figured out the "b" terms, and the initial "a" term was easy, but here is where I have a few doubts; hence, the "unknown." For values when the b term is not zero, I found the integration by parts to be

[tex] \frac{1}{\pi}([\frac{x}{n}sinnx]^{\pi}_{0} + \frac{1}{n}\int^{\pi}_{0}sinnx dx)[/tex] (1)

I know that the integral will come out to give

[tex]2(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...)[/tex]

and the first term in the parts equation (1) will always be zero

So, then I think the answer will be

[tex] f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + \frac{2}{\pi}(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...) [/tex]

But, it just seems a little strange that a_n terms go by squares. Fourier analysis is new to me, and so I can't really gauge how certain functions look like.

Also, the other thing I wanted to double check on was how the function looked; this is a sawtooth, right?

P.S.

Does anybody know the piecewise function command for LaTeX? I thought it was \cases.

P.S.S

Thanks, hopefully the formatting is better now.

[tex]f(x) = \left\{\begin{array} {c}0 \ \ \mbox{for} \ \ - \pi <x<0 & x \ \ \mbox{for} \ \ 0<x<\pi[/tex]

Here is how far I have gotten, using that

[tex]a_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xcosnx dx[/tex]

and

[tex]b_n = \frac{1}{\pi} \int_{- \pi}^{\pi} xsinnx dx[/tex]

arriving at

[tex] f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + (unknown) [/tex]

So, I figured out the "b" terms, and the initial "a" term was easy, but here is where I have a few doubts; hence, the "unknown." For values when the b term is not zero, I found the integration by parts to be

[tex] \frac{1}{\pi}([\frac{x}{n}sinnx]^{\pi}_{0} + \frac{1}{n}\int^{\pi}_{0}sinnx dx)[/tex] (1)

I know that the integral will come out to give

[tex]2(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...)[/tex]

and the first term in the parts equation (1) will always be zero

So, then I think the answer will be

[tex] f(x) = \frac{\pi}{4} + (sinx - \frac{sin2x}{2} + \frac{sin3x}{3}- ...) + \frac{2}{\pi}(cosx+\frac{cos3x}{3^2} + \frac{cos5x}{5^2} + ...) [/tex]

But, it just seems a little strange that a_n terms go by squares. Fourier analysis is new to me, and so I can't really gauge how certain functions look like.

Also, the other thing I wanted to double check on was how the function looked; this is a sawtooth, right?

P.S.

Does anybody know the piecewise function command for LaTeX? I thought it was \cases.

P.S.S

Thanks, hopefully the formatting is better now.

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