Identity theoremFernando Sanz Gamiz
By definition of accumulation point, is closed. To see that it is also open, let , choose an open ball and write . Now , and hence either has a zero of order at (for some ), or else for all . In the former case, there is a function g analytic on such that , with . By continuity of , for all sufficiently close to , and consequently is an isolated point of . But then , contradicting our assumption. Thus, it must be the case that for all n, so that on . Consequently, , proving that is open in . ∎
Theorem 1 (Identity theorem).
Let be a open connected subset of (i.e., a domain). If and are analytic on and has an accumulation point in , then on .
We have that has an accumulation point, hence, according to the previous lemma, it is open and closed (also called ”clopen”). But, as is connected, the only closed and open subset at once is itself, therefore , i.e., on . ∎
|Date of creation||2013-03-22 17:10:38|
|Last modified on||2013-03-22 17:10:38|
|Last modified by||fernsanz (8869)|