Well, what are the definitions of , , etc.?
The following is a worked example:
In the following, for example
What I don't understand is that why they got .
Since
And
But what do we need to substitute for ? Because this function is piece-wise, how do I know which one of the two functions I have to put into the formula?
Those are inner products. Since and are continious functions on the given interval then their inner product is defined by
I belive that's the definition. The resulting polynomial we get in this problem is called "Legendre's polynomial", but unfortunently there is not much about it in my textbook.