Pointwise convergence of holomorphic functions

This can be done by choosing a sequence of polynomials that converges pointwise to a function with a simple pole at a given point. This shows that the limit function is not holomorphic at that point, but it is still holomorphic on a dense, open subset of \Omega.In summary, the theorem states that a sequence of holomorphic functions on a domain \Omega will converge pointwise to a function f that is holomorphic on a dense, open subset of \Omega. However, it can be difficult to find a sequence for which the limit function is not holomorphic on the entire \Omega. One way to do so is by using Runge's theorem and choosing a sequence of polynomials that converges pointwise to a function
  • #1
Radiator1
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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 \(\displaystyle \Omega\).
 
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  • #2
Radiator said:
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.
 

FAQ: Pointwise convergence of holomorphic functions

What is pointwise convergence of holomorphic functions?

Pointwise convergence of holomorphic functions refers to the behavior of a sequence of holomorphic functions as the sequence approaches a given point. It measures how closely the functions in the sequence approximate the limit function at that point.

How is pointwise convergence different from uniform convergence?

Uniform convergence is a stronger form of convergence than pointwise convergence. While pointwise convergence measures the behavior of a sequence of functions at a given point, uniform convergence measures the behavior of the sequence as a whole over the entire domain. Uniform convergence also ensures that the limit function is continuous, whereas pointwise convergence does not.

What is the importance of pointwise convergence in complex analysis?

Pointwise convergence is important in complex analysis because it allows us to study the behavior of a sequence of holomorphic functions at a given point. It is a fundamental concept in the study of power series, which are essential tools in complex analysis.

Can pointwise convergence fail to imply uniform convergence?

Yes, pointwise convergence can fail to imply uniform convergence. This can occur when the domain is unbounded or when the limit function is not continuous. In these cases, the sequence of functions may converge pointwise but not uniformly.

What are some applications of pointwise convergence of holomorphic functions?

Pointwise convergence of holomorphic functions has applications in many areas, including complex analysis, number theory, and physics. It is used to study series expansions and to prove theorems about the behavior of complex functions. It also has applications in numerical analysis and approximation theory.

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