Thread: Periodic functions and Fourier Series

1. Periodic functions and Fourier Series

I'm looking at some examples that I have and I follow them until I get to the result where something like -2*(-1 + cos(Pi*m)) / Pi*m^2
is simplified to -2/Pi*m^2 ((-1)^m - 1)

Why does the cos(Pi*m) turn into (-1)^m?
And if this was sin instead what would happen?

I may be forgetting some old simple rule that I should have learnt

2. Originally Posted by Unoticed
I'm looking at some examples that I have and I follow them until I get to the result where something like -2*(-1 + cos(Pi*m)) / Pi*m^2
is simplified to -2/Pi*m^2 ((-1)^m - 1)

Why does the cos(Pi*m) turn into (-1)^m?
And if this was sin instead what would happen?

I may be forgetting some old simple rule that I should have learnt
$\displaystyle \cos 0 = 1, \, \cos \pi = -1, \, \cos 2\pi = 1, \, \cos 3 \pi = -1, \, ....$ See the pattern?

$\displaystyle \sin m \pi = 0$ as your old forgotten friend the unit circle will gladly tell you if you catch up for a chat ......

3. lol thanks, I realised shortly after I made the post.