# Thread: Sine and Cosine Fourier series

1. ## Sine and Cosine Fourier series

Hi!
Im asked to write sine AND cosine series of $\displaystyle sin(x) ; x \in <0,\pi)$

*what does that mean? i thought the fourier series can exist for the sine part being 0 OR the cosine being 0 OR both sine and cosine being included in the formula.. how do i do then one SINE and another COSINE serie of the same function?

*Im confused with the condition: first of all it doesn't look like a periodic function to me, but im supposed to do a fourier transform on it...

if i imagine it's a periodic function(somehow) then:
*how does the function look like? is it the positive parts of sine one next to the other, thus with period = $\displaystyle \pi$ ... or is it the complete sine with a constant 0 put in between the positive parts thus the period would be $\displaystyle 2\pi$?

*Also, how do i know what the function looks like on x < 0 ? (considering it is periodic, is the function even or odd or none of these?

Thanks!

2. Probably is requested the Fourier expansion of the output of a 'full wave rectifier' ...

$\displaystyle f(x)= |\sin x|$ , $\displaystyle - \pi < x < \pi$ (1)

The function is 'even' so that there are only the terms in 'cosine' and is...

$\displaystyle \displaystyle f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n}\ \cos nx$ (2)

... where...

$\displaystyle \displaystyle a_{0} = \frac{2}{\pi}\ \int_{0}^{\pi} \sin x \ dx = \frac{4}{\pi}$

$\displaystyle \displaystyle a_{n} = \frac{2}{\pi}\ \int_{0}^{\pi} \sin x \ \cos nx \ dx = \left\{\begin{array}{ll} - \frac{4}{\pi\ (n^{2}-1)} ,\,\, n \ even\\{}\\0 ,\,\, n \ odd \end{array}\right.$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. It definitely makes much more sense now... so i guess the question to find the sine fourier form would lead to $\displaystyle b_k = 0$ which would be the result - silly.
Thank you chisigma

4. No, the sine Fourier series for sin(x) would have all $\displaystyle b_k= 0$ except for k= 1: $\displaystyle b_1= 1$ because the sine Fourier sine series for sin(x) is precisely "sin(x)".