Essential singularity

[asi123]

1. The problem statement, all variables and given/known data

Hey guys.
I need to show that this function has an essential singularity at z=0.
I used Taylor series to get what I got, which is a series inside a series...:confused:
And I can't see how am I suppose to show it from here.
Any ideas guys?

Thanks.

2. Relevant equations



3. The attempt at a solution

[statdad]

Without seeing the result, or the original function, it's difficult to say what is right and what isn't.

[asi123]

Without seeing the result, or the original function, it's difficult to say what is right and what isn't.

Sorry :smile:

[Mark44]

Did you mean to write f(z) = cos(e^(1/2))? If so that's a constant function.

I'm having a hard time reading your writing, as your e looks like a cross between an e and a u. Is that thing in the numerator of the exponent on e the digit 1?

[HallsofIvy]

I think it is e^{1/z} not e^{1/2}.

Yes, write out the Taylor's series for ex and replace x with z-1 as you have done. Now, what is the definition of "essential singularity"?

[asi123]

I think it is e^{1/z} not e^{1/2}.

Yes, write out the Taylor's series for ex and replace x with z-1 as you have done. Now, what is the definition of "essential singularity"?

The Laurent series of f(x) at the point a has infinitely many negative degree terms, the thing is, how can you see that trough this series inside a series?

Thanks.

[HallsofIvy]

Oh, you have cos(e1/z). I was only looking at your first e1/z.

Well, e1/z already has an infinite number of negative exponents. Certainly one of the coefficients will cancel out.