Fourier Series of modulus[t] - example of this?

[ZedCar]

Homework Statement

Does anyone know of a website, or a book, where I can see a worked example of the Fourier Series

f(t) = [t]
-∏<t<∏
T=2∏

Finding a0 and an

Of course, it doesn't have to be t, it could be x or any other variable.

Thank you.

Homework Equations

The Attempt at a Solution

 

[ZedCar]

I've found an example here;
http://www.exampleproblems.com/wiki/index.php/FS1

Why is it that in my notes 'an' = -4/[∏(k^2)]

 

[OldEngr63]

You have defined the function as
f(t) = [t]
-∏<t<∏
T=2∏
but that is not the example that you show worked out. The example that you show worked out is this one
f(t) = |t|
-∏<t<∏
T=2∏
This difference being that, in the latter case, the function between - π and +π is the absolute value of t; in the original problem statement, as you gave it, [t] is not really a well defined function.

The absolute value function is an even function, so when we proceed to evaluate the coefficients for the Fourier series, the first one works out as

a_o = \frac{1}{2π}∫_-π^π f(t) dt
= \frac{1}{π}∫₀^π t dt
=\frac{π}{2}

The others follow in the usual fashion with the cosine multiplication.

 

[vela]

I've found an example here;
http://www.exampleproblems.com/wiki/index.php/FS1

Why is it that in my notes 'an' = -4/[∏(k^2)]

That should be either n or k, not both, in your equation. Let's just use n.

What does the result of the integration give you when you assume n is odd and n is even?