Is there an easy way to find the weight to make eigenfunctions orthogonal? (by hand and with an "easy" logic process, no numerical method type stuff)
Let's move on to an example taken from solving Laplace's equation on a ring sector, at some point we get (with $E_n$ being constants):
Is there an easy way to find that those functions are orthogonal with weight functions $\frac{1}{r}$?
So that we can do:
Thus allowing us to go further in the solving.
So:
- Main question: is there a way to find the weight function of this example? (assuming a case where we have to find it and we don't know it).
- If so, is there anything similar that can be applied to find the weight function in general? or in some specific useful cases? (If that last question covers too much, it can be ignored).