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Real and complex behaviour of the base change function (was: The "cheta" function)
#31
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.
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RE: singularities of base change eta -> e - by bo198214 - 08/20/2009, 10:28 AM

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