Calculating Fourier Series of f(x) = |x| - \pi

[squenshl]

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

 

[LCKurtz]

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.

 

[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?

 

[squenshl]

I see. Since f(x) is an even function, when it goes to finding b_n 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

 

[LCKurtz]

I see. Since f(x) is an even function, when it goes to finding b_n 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 a_n 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.

 

[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).