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Question: Calculate the Fourier series of f (x) = e^x on the interval −π ≤ x ≤ π.

I am new to Fourier Series. I managed to find a0 and am. However, I have no idea where does the second am comes from(see solution attached). Could someone please explain what is happening in the second am?

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Tommy_Smith
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  • I see what's going on here. If you look at the equation just above the one with =>, $a_m$ also appears on the right hand side. If you solve that equation for $a_m$, you get the one with =>. – thang Feb 17 '15 at 21:56
  • @thang Thank you such much. The question was much sillier than I thought it was. – Tommy_Smith Feb 17 '15 at 22:01

1 Answers1

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First equality: Use the Definition of the Fourier Expansion coefficients, here $a_m$ are the cosine coefficients.

2nd equality: Integration by parts where the cosine is integrated.

3rd equality: Again it is integrated by parts; the sine function is integrated here.

4th equality: Because you have the Expression $\int_{- \pi}^\pi e^x cos(mx)dx = a_m$ again.

You can make some Manipulation with your equation; Hint: compare first line for $a_m$ with the 4th line. Then you obtained the desired result.

kryomaxim
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  • Thank you for your help and I understand the "=> am" line now. But then could you please explain how to obtain the final line of the solution? Can I get the last line with a0? Or I have to expand all the items and subbing in pie and minus pie? – Tommy_Smith Feb 17 '15 at 22:07
  • The last line in $a_m$ is only the Substitution of the Integration bounds with $sin(m \pi)=0,cos(m \pi)=(-1)^m)$ for natural numbers $m$. It holds $a_0 = [\frac{1}{\pi}e^x]_{-\pi}^\pi= \frac{1}{\pi}e^\pi - \frac{1}{\pi}e^{-\pi}$. – kryomaxim Feb 17 '15 at 22:11
  • But then what's the point of calculate a0 here if we dont need it in further calculation? – Tommy_Smith Feb 17 '15 at 22:18
  • The quantity $a_0$ is needed to obtain the full Fourier series Expansion: $f(x) = a_0 + \sum_{m=1}^\infty a_m cos(mx) + Expansioninsin(mx)$. – kryomaxim Feb 17 '15 at 22:23