Analytic Extension on a Complex Function

In summary, the problem is to prove that there cannot be an analytic extension of the function f(x) = Series on n from 1 to infinity: x^(n!), containing the unit disk, due to the uniqueness of extensions theorem and the fact that the roots of unity are dense along the unit circle. The professor's hint suggests showing that the function diverges as you approach a dense set of points on the unit circle, which would also mean that it diverges at the roots of unity when approaching from within the unit disk.
  • #1
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Homework Statement


Prove there cannot be an analytic extension containing the unit disk of:

f(x) = Series on n from 1 to infinity: x^(n!)


Homework Equations



Unique Extension theorem, no real explicit equations I can think of.


The Attempt at a Solution



So far I've proved the range of convergence is 1 and that it diverges for all |z|=1. There is an accumilation point for the coincidence set of f(x) which is all |z|=1 thus the extension must be unique. Since the extension must contain the unit disk it follows that all z such that |z|=1 must be defined on the extension in which it agrees with f(x) for all |z|<1. My professor gave us the hint to show that the roots of unity are dense along the unit circle... though I'm unsure how to apply this to the problem itself. I was hoping one of you could help me or give me some direction on the problem.

I'm guessing I a)Need to show that the extension cannot be defined on a root of unity without a contradiction (No clue how to show this). b)Since the roots of unity are dense along the unit circle its impossible for any domain containing the unit circle to not be defined on a root of unity, thus the extension cannot exist.

It seems like this is the steps I need to take although I'm unsure how to prove or start proving them. I supposed for a, I can just state F(e^(2ipim/n) = k for some nth root of unity, but I don't know how to derive a contradiction from that.

Thanks for any help you guys can provide!
 
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  • #2
Judging by the hint he gave, I think he is looking for a proof along the lines of showing that the function diverges as you approach a dense set of points on the unit circle. For example, show it diverges as r->1 for r real. If that's clear then think about what what happens to f(r*u) where u is a root of unity as r->1.
 

What is analytic extension on a complex function?

Analytic extension on a complex function is the process of extending a function, defined on a subset of the complex plane, to a larger domain in the complex plane while maintaining its analyticity. This allows for the function to be evaluated at points outside of its original domain.

Why is analytic extension useful?

Analytic extension is useful because it allows for the evaluation of a function at points where it may not be defined, providing a more complete understanding of the function's behavior. It also allows for the application of mathematical techniques, such as integration and differentiation, to the extended function.

What are the main techniques used for analytic extension?

The main techniques used for analytic extension include Cauchy's integral formula, Taylor series expansion, and Laurent series expansion. These techniques use the properties of analytic functions to extend the domain of a function while preserving its analyticity.

What is the relationship between analytic extension and analytic continuation?

Analytic extension and analytic continuation are closely related concepts, with the main difference being the direction in which the extension is performed. Analytic extension extends a function from a smaller domain to a larger one, while analytic continuation extends a function from a larger domain to a smaller one. Both techniques utilize the properties of analytic functions to extend the domain of a function while maintaining its analyticity.

Are there any limitations to analytic extension?

Yes, there are some limitations to analytic extension. It is not always possible to extend a function analytically, as there may be singularities or branch points in the original domain that cannot be extended. Additionally, the extended function may not be unique, as there can be multiple ways to extend a function to a larger domain.

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