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

• squenshl
In summary, the conversation discussed how to find the Fourier series of the function f(x) = |x| - \pi on the interval (\pi, \pi]. It was determined that the function is even, which allows for the use of half-range expansions and symmetry when finding coefficients. It was also noted that the Dirichlet conditions should be considered when determining the convergence of the Fourier series for this function.

#### squenshl

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

squenshl said:
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?

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.

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

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

squenshl said:
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.

Cheers. Got it, that was easy after all.

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

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).