# Essential singularity

1. Feb 13, 2009

### 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...
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

2. Feb 13, 2009

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

3. Feb 13, 2009

### asi123

Sorry

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4. Feb 13, 2009

### Staff: Mentor

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?

5. Feb 13, 2009

### HallsofIvy

Staff Emeritus
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"?

6. Feb 14, 2009

### asi123

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.

7. Feb 14, 2009

### HallsofIvy

Staff Emeritus
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.