Pointwise convergence of holomorphic functions

[Radiator1]

Hello.

In my complex analysis book I've read a theorem which says that if a sequence \(\displaystyle \{ f_n \}\) of holomorphic functions on a domain \(\displaystyle \Omega\) converges pointwise to a function \(\displaystyle f\), then \(\displaystyle f \) is holomorphic on a dense, open subset of \(\displaystyle \Omega\).

I know how to prove this theorem. I just find it hard to come up with an example of a sequence (as described above) for which the limit $f$ is not holomorphic on the entire \(\displaystyle \Omega\).

 

[Euge]

Hello.

In my complex analysis book I've read a theorem which says that if a sequence \(\displaystyle \{ f_n \}\) of holomorphic functions on a domain \(\displaystyle \Omega\) converges pointwise to a function \(\displaystyle f\), then \(\displaystyle f \) is holomorphic on a dense, open subset of \(\displaystyle \Omega\).

I know how to prove this theorem. I just find it hard to come up with an example of a sequence (as described above) for which the limit $f$ is not holomorphic on the entire \(\displaystyle \Omega\).

Hi Radiator,

By application of Runge's theorem, you should be able to find a sequence of polynomials whose pointwise limit has a point of discontinuity.