Fourier Series

**atwinix** · August 17th 2007, 12:56 AM

Hello,

Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.

**CaptainBlack** · August 17th 2007, 11:02 AM

Originally Posted by atwinix

Hello,

Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.

What is the problem with these? The third line in each tells you that the function is periodic with period
$2 \pi$ so any interval of that length is what you use in the integrations and the basis functions are
$\sin(nx)$ and $\cos(nx)$ .

RonL

**atwinix** · August 17th 2007, 08:29 PM

Yes, I know how the Fourier series expansion work.

I am having problem with the integration between the limits for question 6 (iv) as you get something like:

∫ (π+t) cos (nt) dt + ∫ (π-t) cos (nt) dt

**CaptainBlack** · August 17th 2007, 09:20 PM

Originally Posted by atwinix

Yes, I know how the Fourier series expansion work.

I am having problem with the integration between the limits for question 6 (iv) as you get something like:

∫ (π+t) cos (nt) dt + ∫ (π-t) cos (nt) dt

Using integration by parts with $u=(\pi-t)$ and $dv=\cos(nt)$ :

$\int_0^\pi (\pi-t) \cos(nt)~dt = \left[ \frac{(\pi-t)}{n} \sin(nt) \right]_0^{\pi}+\int_0^{\pi}\frac{1}{n^2} \sin(nt)~dt$ $=\frac{1}{n^2}(1-(-1)^n)$

which is $2/n^2,\ n=1, 3, 5, ..$ and $0,\ n=2,4,..$

RonL

**atwinix** · August 18th 2007, 01:13 AM

Oh!!! So that's the catch! I shall try it.

Thanks loads.

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