Would someone be kind enough to explain to me what would be the Fourier series of $f(x)=1$ and $f(x)=x$ on $[0,1]$?
See, all the equations I can find are for intervals of the type $[-L,L]$. Now I don't know what $f$ is on $[-1,0]$, so I can't tell if it's even or odd, nor how to use the formula given here, since I am not sure whether I can use $L=1$ as my interval isn't symmetric about 0.
Just need an short explanation for $f(x)=1$ and $f(x)=x$ so I can figure out the right way to use those equations. Also, I am asked to use series that contain only sine terms and only cosine terms. Does that mean I need to apply some kind of restriction or will it just come out that way regardless? Thanks.
Think f(x) as a piecewise defined function where:
$$ f(x) = \left\{ \begin{array}{l l} 1 & \text{if }x \in [0,1] \\ 0 & \text{else} \end{array} \right. $$
In that case L, which is half the period by the way, is 1. As an example, here I'll write $a_m$:
$$ a_m = \frac{1}{1} \left[ \int_{-1}^{0} 0 \cdot cos\left(\frac{m\pi x}{1}\right)\, dx + \int_{0}^{1} 1 \cdot cos\left(\frac{m\pi x}{1}\right)\, dx \right] = \\ =\int_{0}^{1}cos\left(\frac{m\pi x}{1}\right)\, dx $$
Similarly, you can calculate $b_n$ and get the complete serie.
Same thing with $f(x)=x$