Fourier series sketching

Question

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ([-2pi,2pi]in this case). If the function given is odd function, we extend it by reflecting the thing we sketched corresponds to the boarder line (in this case the lines are x=0, x=pi, x=-pi); if it is an even function, we copy the sketching to those region without reflecting.

Sorry for my bad English.

Answer 1

The Fourier series converges to a certain extension of the given function.

1. cosine series on $[0,L]$: even periodic extension
2. sine series on $[0,L]$: odd periodic extension
3. full Fourier series (sines and cosines on $[-L,L]$): periodic extension.

To obtain the required extension,

1. Define $f(x)=f(-x)$ on $[-L,0)$ and proceed to 3.
2. Define $f(x)=-f(-x)$ on $[-L,0)$ and proceed to 3.
3. Extend periodically by $f(x+2L)=f(x)$.

Answer 2

There is no ambiguity on the function definition. It is defined as $x-\pi$ on $(0,\pi]$ and even with period $2\pi$, so that it is $-x-\pi$ on $(-\pi,0]$.

Anyway, the Fourier series is to be computed on the range $(-2\pi,2\pi]$, there will be no $2\pi$ harmonic.