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 |
---|---|
Canonical name | ZeroesOfAnalyticFunctionsAreIsolated |
Date of creation | 2013-03-22 13:38:10 |
Last modified on | 2013-03-22 13:38:10 |
Owner | brianbirgen (2180) |
Last modified by | brianbirgen (2180) |
Numerical id | 8 |
Author | brianbirgen (2180) |
Entry type | Result |
Classification | msc 30C15 |
Synonym | zeros of analytic functions are isolated |
Related topic | Complex |
Related topic | LeastAndGreatestZero |
Related topic | IdentityTheorem |
Related topic | WhenAllSingularitiesArePoles |