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
Thanks in advance!
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
Thanks in advance!
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 $\displaystyle f(x)$ is an even function, then $\displaystyle b_n = 0$, i.e $\displaystyle f(x) \approx \frac{a_0}{2} + \sum a_n \, \cos nx$
If $\displaystyle f(x)$ is an odd function, then $\displaystyle a_n = 0$, i.e $\displaystyle f(x) \approx \frac{a_0}{2} + \sum b_n \, \sin nx$
You need to know the orthogonal properties of sine and cosine.
(If L>0 we have):
$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m$.
$\displaystyle \int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}$.
$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0$.
Here $\displaystyle L=\pi$.