08/20/2009, 10:28 AM

Though currently I wonder whether these arbitrary close singularities indeed imply that the function is not analytic in any point.

I mean there is a theorem that if a holomorphic function sequence converges locally uniformly (i.e. for each point there is a neighborhood where it converges uniformly) then the limit is again a holomorphic function (which is not true for just differentiable functions).

However I dont think that the inverse statement is also true, that if a function sequence does not converge locally uniformly that then resulting function can not be holomorphic.

For example a sequence of non-continuous functions can have a continuous function as a limit. Also Jay showed that the singularities gets milder with increasing n. So there maybe a very little tiny hope that the resulting function is analytic despite.

I mean there is a theorem that if a holomorphic function sequence converges locally uniformly (i.e. for each point there is a neighborhood where it converges uniformly) then the limit is again a holomorphic function (which is not true for just differentiable functions).

However I dont think that the inverse statement is also true, that if a function sequence does not converge locally uniformly that then resulting function can not be holomorphic.

For example a sequence of non-continuous functions can have a continuous function as a limit. Also Jay showed that the singularities gets milder with increasing n. So there maybe a very little tiny hope that the resulting function is analytic despite.