## Finding the function given a Fourier Series.

1. The problem statement, all variables and given/known data
From the Fourier series

$$\frac{1}{2} - \frac{1}{4}cos(x) + \sum\frac{(-1)^{n}}{1-n^{2}}cos(nx)$$

of

(1/2)x*sinx on $$[-\pi,\pi]$$, find the function whose Fourier Series on $$[-\pi,\pi]$$ is

$$\frac{3}{4}sin(x) - \sum\frac{(-1)^{n}}{n-n^{3}}sin(nx)}$$

Both sums go from n=2 to n=infinity. The latex stack was showing up weird.
2. Relevant equations

3. The attempt at a solution
Well, this seems like a fair bit of intuition. I tried to plug in F-Series formula for sin(x) and cos(x) into both F-Series but didn't get anywhere. I then realized that the second F-Series bears some resemblance to the integration of the first series. For example, the indefinite integral of the sum in the first F-Series is exactly the sum in the second series, aside from a constant. The integral of the -(1/4)cos(x) term yields (1/4)sinx, which is close and hence here has been everything I've tried.

Any suggestions to get me going in the right direction?

I also tried setting the a_n coefficient from the first F-Series equal to n*b_n of the second, and then I had the integrals equated but I can't really do much from there. Differentiating both integrals just gives 0= 0.

EDIT: Sorry, I was missing "sin(x)" in my second F-Series Formula.
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 So no one has any ideas? BTW this is a bump because I made a huge mistake on the latex for second F-Series.
 well it looks to me like you could consider the fourier series to the the resultant waveform whn a pure wave form is applied as the x in a f(x). Imagine that that f(x)=x^2. Then the pure waveform is modulated by f(x) so that where we had sin(x) we now have (sin(x)). Now we apply the appropriate trig identity: (sin(x))^2 = (sin(x))*(sin(x))= sin(a)*sin(B) where A=B (just so you recognise it!) sin(A)*sin(B)=(1/2)(cos(A+B)-cos(A-B)) (sin(x))^2=(1/2)*(-cos(2*x)+cos(0)) (look out for that negative on the first cos!) The fourier series you quote is likely, I believe, to have resulted from such considerations at a higher power than 2. Such functions arise for instance when a small varying current passes through a silicon diode; the output voltage may well be represented by a function like this.

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