Fourier Series involving modulus function

[nicholasng]

Hi all,

I've come across this problem as stated:

Let f(x) be the function f(x) = |x| + π, if -π < x < π

Find the first two non-zero terms of the Fourier Series for f(x) in exact value.

I understand that it is an even function given that f(-x) = |-x| + π = |x| + π = f(x), thus b_n = 0 for all n=1,2,3...

However, as I got stuck and referred to suggested solution, in the steps the π was omitted from f(x), leaving only f(x) = |x| in the sense that when finding a₀, the solution took |x| as f(x) instead of |x| + π.

May I know the reason behind that?

Thank you.

 

[skaa]

\(\displaystyle a_n=\frac{1}{\pi}\int_{-\pi}^\pi (|x|+\pi)\cos(nx)dx=\frac{1}{\pi}(\int_{-\pi}^\pi |x|\cos(nx)dx+\pi\int_{-\pi}^\pi\cos(nx)dx)\)
The second item is 0 for any \(\displaystyle n\) because \(\displaystyle 2\pi\) is a period for \(\displaystyle \cos(nx)\) on which its integral is 0. (Draw its graph, and you will see this.)