If and we proved that
- Prove the above formula for is valid for any .
- If when does the above formula hold?
- Proof. We write
- The problem with this formula if is that is not defined there. I’m not sure what else Apostol wants us to say other than we can not meaningfully write if .
On b) taking the absolute value of the logarithm you extend the definition for every x different than zero. He for x equal zero we have f of zero equals zero for every r
I added that the formula is only valid for rational r if x≤0, since this has already been proven by other means earlier in the book.
It’s not valid for all rational numbers though. It works only for , where is odd.