zeroes of analytic functions are isolated
The zeroes of a non-constant analytic function on are isolated. Let be an analytic function defined in some domain and let for some . Because is analytic, there is a Taylor series expansion for around which converges on an open disk . Write it as , with and ( is the first non-zero term). One can factor the series so that and define so that . Observe that is analytic on .
To show that is an isolated zero of , we must find so that is non-zero on . It is enough to find so that is non-zero on by the relation . Because is analytic, it is continuous at . Notice that , so there exists an so that for all with it follows that . This implies that is non-zero in this set.
|Title||zeroes of analytic functions are isolated|
|Date of creation||2013-03-22 13:38:10|
|Last modified on||2013-03-22 13:38:10|
|Last modified by||brianbirgen (2180)|
|Synonym||zeros of analytic functions are isolated|