Analyzing the Singularity of f(z) = z cos(1/z) Using Solutions to f(z) = 0

[Gwinterz]

Hey,

I have been asked to show that z = 0 is not a removable singularity of:

f(z) = z cos(1/z)

The catch is, I have to show this by first finding all the solutions to the equation (f(z) = 0) then use them to show that the singularity is not removable.

The only relevant theorem I can find is the "Isolated zeros theorem", but that is related to showing a function is analytic, I am not sure how it could apply hear.

Is there a theorem which links the solutions to an equation to the singularities of the equation?

I noticed that the solutions to cos(1/z) tend to zero as 'k' tends to infinity, but I am not sure if that is useful.

 

[FactChecker]

Suppose that there is an extension, e(z), of the function that is analytic at z=0. Then can you find a set of zero points of e(z) that have a cluster point at z=0? Then what would the isolated zeros theorem tell you?

 

[Gwinterz]

It would tell me that the function is not analytic at the point z = 0, as there is more than one zero.

Is that right?

Does that mean that the singularity is not isolated?

 

[FactChecker]

It would tell me that the function is not analytic at the point z = 0, as there is more than one zero.

Not just more than one, there are infinitely many around z=0. The only analytic function with that property is the constant function, f(z)=0.

 

[Gwinterz]

So because the only analytic function is f(z) = 0, does that mean our f(z) can't be analytic? Then since it isn't analytic, it breaks the definition of an isolated singularity?

 

[FactChecker]

For a homework exersize, you should work out the details and complete the proof. I think you have gotten enough hints to complete it rigorously.

 

[Gwinterz]

My problem is with the details, I think once I understand the idea I can begin to formulate the proof.

At the moment I really don't see the relationship between the solutions and the singularity.

These questions are really tough for me and I waste so much time trying to figure out what set of theorems to apply then realizing it doesn't work.

I get that the solutions tend to zero and that there is a consequence to that. But I don't understand the relevance of f(x) = 0. Was what I said above the right idea? It was a pretty big hunch for me.

Using the Laurent series it's clear the singularity is essential. But according to my hunch above I would be working towards it being not isolated. Doesn't it have to be isolated to be essential?

I keep getting stuck I'm these loops of thinking.

 

[FactChecker]

You want to reference the isolated zero theorem to say that z=0 is an essential not a removable singularity. Do that by showing that the zero at z=0 is not isolated, so there can not be an analytic extension of $z\cos(1/z)$.

Correction: the singularity is not an essential singularity because it is not isolated.

 

[Gwinterz]

Thanks for your answer and patience.

I am just wondering about one last thing.

When I see essential singularities defined, they are always a subset of isolated singularities.

Is it correct to say that essential singularities must be isolated?

I'm just a bit confused here because if that's true, wouldn't showing it is not isolated imply it isn't essential?

 

[FactChecker]

When I see essential singularities defined, they are always a subset of isolated singularities.

Is it correct to say that essential singularities must be isolated?

I'm just a bit confused here because if that's true, wouldn't showing it is not isolated imply it isn't essential?

Good catch. I stand corrected. It is not an essential singularity because it is not isolated. You can still say that it is not a removable singularity. I have corrected my post #8.

 

[Gwinterz]

Awesome,

Thanks so much!

I think I got it!