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Struggle is an understatement!

I'm trying to get my head around Fourier analysis and I have the equation : $$f(x)=2\pi^2+6x^2$$ unfortunately I have no idea where to start and my coursework depends on this as the actual calculation is worth 10 marks.

Please any help would be appreciated.

user88595
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user130732
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    What's the question exactly? Find a Fourier series to that expression? – user88595 Feb 22 '14 at 15:47
  • Might example 1 on page 8 be of any help? http://math.bard.edu/~belk/math461/FourierSeries.pdf – user88595 Feb 22 '14 at 15:49
  • what have you tried and what do you know about Fourier series / transforms? Assuming it's Fourier series, one thing to consider is whether the function is odd or even. If it's odd you only need to use sines. If it's even you only need cosines. Otherwise you need both. – TooTone Feb 22 '14 at 15:49
  • i have basically just started to learn this last week so hence the confusion. the function is even that i know. I am trying to find the Fourier series to the expression. I seem to get so far and am totally confused so any help would be appreciated. I was able to calculate f(x)=x^2 but because the question has changed i am totally lost in breaking down the equation. But i will go look at your link see if that helps. – user130732 Feb 22 '14 at 16:20
  • To compute a Fourier series you need a period length. And you need to tell over which fundamental period your expression describes the desired periodic function. – Lutz Lehmann Feb 22 '14 at 16:35
  • @user130732 You ought to edit your question to say how far you have actually gotten before you get confused, and then you can get some targeted help here. Scan/photograph your work if necessary. Also, as the previous comment says, you do need to know what period your series is going to be over. – TooTone Feb 22 '14 at 16:43
  • @user130732, you should be way clearer in your question and ask explicitly what is the Fourier series of the function $;f(x);$, otherwise people begins to guess. By the way, on what interval?? – DonAntonio Feb 22 '14 at 16:43
  • sorry for the confusion, I am new to this. So I guess what i was trying to ask is what is the Fourier series of the function f(x)=2π²+6x² and the period length is 2π. i am away to investigate my notes and i will scan what i have done so far. sorry again for confusion – user130732 Feb 22 '14 at 19:16
  • Use also the linearity of the Fourier transform to reuse your previous results. The Fourier series of a constant is trivial. – Lutz Lehmann Feb 22 '14 at 19:51

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The Fourier transforms are all linear transforms, if the given function is a sum you can treat each addens separately, if the given function contains constant factors you can keep them out of the calculation and multiply them back in the result.

Obviously the Fourier series of the constant $1$ is the constant $1$, i.e., all coefficients except the constant one are zero.

For $x^2$ the Fourier coefficients for the symmetric fundamental perion $[-L,L]$ with basis functions $e^{i\frac\pi L kx}$ are \begin{align} c_K=\frac1{2L}\int_{-L}^Lx^2\,e^{-i\frac\pi L kx}\,dx =\frac{L^2}2\int_{-1}^1 x^2\,e^{-i\pi kx}\,dx\\ \end{align} which can be solved using partial integration reducing the degree of the factor $x^2$.

Lutz Lehmann
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