We covered poles in the page on singularities. We now discuss zeroes.
Zeroes
If is analytic at , and , then we say that has a zero of order at .
We have an additional proposition. If , with and both analytic at , and , and has a zero of order at , then has a pole of order at . Then, if has a zero of order at and has a zero of order at , then has a:
- Pole of order if
- Neither pole nor zero if
- Zero of order if