# Fourier sine series...

• February 10th 2008, 05:05 AM
hunkydory19
Fourier sine series...
Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all (Thinking)

• February 10th 2008, 05:12 AM
kalagota
Quote:

Originally Posted by hunkydory19
Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all (Thinking)

What do you mean by fourier sine? is it the sine part of the Fourier Series?
if does, this should be a good hint:

If $f(x)$ is an even function, then $b_n = 0$, i.e $f(x) \approx \frac{a_0}{2} + \sum a_n \, \cos nx$

If $f(x)$ is an odd function, then $a_n = 0$, i.e $f(x) \approx \frac{a_0}{2} + \sum b_n \, \sin nx$
• February 10th 2008, 06:43 AM
hunkydory19
Thanks kalagota,

So $a_n = 0$

and $b_n = \frac{2}{\pi}\int^\pi_0 cos x sin nx \, \mathrm{d}x$

But how do integrate this, I tried doing it by parts but realised that it will just go on forever!
• February 10th 2008, 07:14 AM
Krizalid
You can turn a product of sine and cosine into a sum, do you know that formula? Applyin' that the rest follows.
• February 10th 2008, 07:48 AM
ThePerfectHacker
You need to know the orthogonal properties of sine and cosine.
(If L>0 we have):

$\int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m$.

$\int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}$.

$\int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0$.

Here $L=\pi$.
• February 10th 2008, 07:53 AM
Jhevon
Quote:

Originally Posted by ThePerfectHacker
You need to know the orthogonal properties of sine and cosine.
(If L>0 we have):

$\int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m$.

$\int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}$.

$\int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0$.

Here $L=\pi$.

indeed! and the first two are 0 if $m \ne n$

you know i didn't know about these rules until after my differential equations class was over! i always had to do it the hard way (the product to sum formulas)