places of holomorphic function
If is a complex constant and a holomorphic function in a domain of , then has in every compact (closed (http://planetmath.org/TopologyOfTheComplexPlane) and bounded (http://planetmath.org/Bounded)) subdomain of at most a finite set of http://planetmath.org/node/9084-places, i.e. the points where , except when in the whole .
Proof. Let be a subdomain of . Suppose that there is an infinite amount of -places of in . By http://planetmath.org/node/2125Bolzano–Weierstrass theorem, these -places have an accumulation point , which belongs to the closed set . Define the constant function such that
for all in . Then is holomorphic in the domain and in an infinite subset of with the accumulation point . Thus in the -places of we have
Consequently, the identity theorem of holomorphic functions implies that
in the whole . Q.E.D.
|Title||places of holomorphic function|
|Date of creation||2013-03-22 18:54:18|
|Last modified on||2013-03-22 18:54:18|
|Last modified by||pahio (2872)|