Define the ball at of radius as and is the closed ball at of radius .
What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always converges to a holomorphic function but that’s another theorem altogether), the function is not identically zero and furthermore the function is not zero on the boundary of some ball, then eventually the functions of the sequence have the same number of zeros inside this ball as does the function.
Do note the requirement for not being identically zero. For example the sequence converges uniformly on compact subsets to , but have no zeros anywhere, while is identically zero.
An immediate consequence of this theorem is this useful corollary.
If is a region and a sequence of holomorphic functions converges uniformly on compact subsets of to a holomorphic function , and furthermore if never vanishes (is not zero for any point in ), then is either identically zero or also never vanishes.
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
|Date of creation||2013-03-22 14:17:55|
|Last modified on||2013-03-22 14:17:55|
|Last modified by||jirka (4157)|