# Proving Essential Singularity at z=0: Using Taylor Series Method

• asi123
In summary, The conversation is about finding an essential singularity at z=0 for a function using a Taylor series. The initial attempt involved using a series inside a series, but there were difficulties in interpreting the results. Suggestions were made to clarify the function and its series, and to consider the definition of an essential singularity. It was determined that the function was actually cos(e^(1/z)) and that e^(1/z) already has infinitely many negative exponents, making it a good candidate for an essential singularity. The conversation concludes with a discussion about how to prove the existence of an essential singularity through the series.
asi123

## Homework Statement

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.

## The Attempt at a Solution

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

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

Sorry

#### Attachments

• scan0004.jpg
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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?

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"?

HallsofIvy said:
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.

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.

## What is an essential singularity?

An essential singularity is a mathematical concept that describes a point in a complex function where the function is undefined or infinite. It is different from a removable singularity, where the function can be made continuous by defining the value at that point.

## How is an essential singularity different from a pole?

An essential singularity is different from a pole in a complex function because it cannot be removed or made continuous by defining a value at that point. A pole, on the other hand, can be removed by defining the value of the function at that point.

## Can essential singularities occur in real-valued functions?

No, essential singularities can only occur in complex-valued functions. This is because only complex functions can have points where the function is undefined or infinite.

## What is the significance of essential singularities in complex analysis?

Essential singularities play an important role in complex analysis because they are crucial for understanding the behavior of complex functions near these points. They also help in determining the analyticity of a function and the presence of other types of singularities.

## How are essential singularities related to the theory of residues?

The theory of residues, which is used to evaluate complex integrals, relies on the presence of essential singularities in the complex function. The residues at these singularities are used to determine the value of the integral, making essential singularities an important concept in this theory.

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