Determine the singularity type of the given function (Theo. Phys)

In summary, the conversation is about a function f(z) = cos(z+1/z) and the existence of a singular point at z=0. The person is unsure how to approach and solve for this and is seeking help and explanation. They mention that the function mainly depends on z+1/z and use a mathematical approach to investigate the behavior of the function at z=0.
  • #1
starstruck_
185
8
Homework Statement
Given the function
f(z) = cos(z+1/z) determine the function's singular point(s), and whether it is a pole, essential singularity, or removable singularity. Determine the residue if it is an essential singularity or a pole.
Relevant Equations
A pole if:
lim f(z) = infinity
z->a
Order of a pole:
Order P that satisfies
lim (z-a)^P * f(z) = finite non-zero number
z->a

Possibly use Taylor expansion or L'Hopital's rule depending on the type of limits obtained
NOTE: Was not sure where to post this as it is a math question, but a part of my "Theoretical Physics" course.

I have no idea where to start this and am probably doing this mathematically incorrect.

given the function f(z) = cos(z+1/z) there should exist a singular point at z=0 as at z = 0; lim as z-> 0 cos(z+1/z) diverges since

lim as z->0 (z+1/z) -> infinity and so cos (z+1/z) oscillates between -1 and 1 (?).

I'm not sure how to do this and I think what I am doing here is definitely wrong, so any help/explanation would be appreciated.
Since the function f(z) depends mainly on z+1/z, that is the part I will deal with:

lim z-> 0 (z + 1/z) = lim z->0 (z) + lim z -> 0 (1/z)

= 0 + lim z-> 0 (1/z)
Where 1/z has a pole at z= 0 of order p =1
so cos(z+1/z) should also have a pole at z-0 of order P =1 ??

I'm really confused, and completely lost on how to approach this/why.
 
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  • #2
Hi.
It is enough to invetigate
[tex]z\ cos(1/z)=\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!z^{2n-1}}[/tex]
 

Related to Determine the singularity type of the given function (Theo. Phys)

1. What is a singularity?

A singularity is a point in a mathematical function where the value of the function becomes infinite or undefined.

2. How do you determine the type of singularity in a function?

The type of singularity in a function can be determined by analyzing the behavior of the function near the singularity point. This can involve taking limits, graphing the function, or using other mathematical techniques.

3. What are the different types of singularities?

There are three main types of singularities: removable, essential, and pole. A removable singularity occurs when the function can be extended to be continuous at the singularity point, an essential singularity occurs when the function has an infinite number of oscillations or is not defined near the singularity point, and a pole singularity occurs when the function approaches infinity near the singularity point.

4. Can a function have more than one singularity?

Yes, a function can have multiple singularities. These singularities can be of the same type or different types.

5. Why is it important to determine the singularity type of a function?

Determining the singularity type of a function is important because it can provide insight into the behavior and properties of the function. It can also help in solving equations and understanding the convergence or divergence of the function.

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