# Fourier series

1. Mar 21, 2010

### squenshl

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

2. Mar 21, 2010

### LCKurtz

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

3. Mar 21, 2010

### 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.

4. Mar 21, 2010

### LCKurtz

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

5. Mar 22, 2010

### 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

6. Mar 22, 2010

### LCKurtz

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.

7. Mar 22, 2010

### squenshl

Cheers. Got it, that was easy after all.

8. Mar 23, 2010

### 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.

9. Mar 23, 2010

### LCKurtz

Draw a couple of periods of f(x). Read what the Dirichlet conditions say about convergence of the FS and apply it to this f(x).