# Fourier series

squenshl
How do work out the Fourier series of f(x) = |x| - $$\pi$$ on ($$\pi$$,$$\pi$$].

Homework Helper
Gold Member
How do work out the Fourier series of f(x) = |x| - $$\pi$$ on ($$\pi$$,$$\pi$$].

What have you tried? Have you noticed that is an even function?

squenshl
I know that |x| = x, x $$\geq$$ 0 & -x, x < 0.
So does that mean |x| - $$\pi$$ = x - $$\pi$$, x $$\geq$$ 0 & -x - $$\pi$$, x < 0.
Then just find the Fourier series as a sum of these functions. If not I don't have a clue where to start.

Homework Helper
Gold Member
Yes, that would work. But the fact that the function is even can reduce the work by half. Have you studied half-range expansions?

squenshl
I see. Since f(x) is an even function, when it goes to finding bn you multiply an even function with sin which an odd function to get an odd function and the integral of an odd function is always zero

Homework Helper
Gold Member
I see. Since f(x) is an even function, when it goes to finding bn you multiply an even function with sin which an odd function to get an odd function and the integral of an odd function is always zero

Yes. And for the an you have even times even and you can use symmetry, which will help you with the absolute values. Again, look at "half range" expansions.

squenshl
Cheers. Got it, that was easy after all.

squenshl
Are there values of x at which this series fails to converge to f(x). To what values does it converge at these points.