Hello, I have some problems understanding what is above on the image.
Firstly, he defines an "odd extension" of any function. I don't really understand what this means, how is it an "odd extension" in any way? And then he says that $F(x) = \sum_{n=1}^\infty b_n \sin (\frac{n\pi x}{L})$ but that is only true if $F$ is odd. We don't know what $F$ is, it is any function so how can he conclude that?
My second question is to do with the example. The function $F(x)$ is neither odd or even, so why is he calculating $b_n$ like that? That definition of $b_n$ is only true if $F$ is odd.
I am very confused