Find the Laurent Expansion of f(z) and Classify Residues

In summary, the conversation is about finding and classifying the residues of the function f(z)=\frac{1}{(z+i)^2(z-i)^2}. The speaker mentions that they factored the function and found a double pole at z=+i and z=-i. They then question if they need to perform a laurent expansion to show that the poles are of order 2 or if stating the power of the brackets is enough. The other person responds that they should do two laurent expansions and that it is not necessary to compute the series to find the poles. They also state that the question is only about finding the residues, so the classification of the poles is not needed. In summary, the conversation suggests that the
  • #1
latentcorpse
1,444
0
Say you have [itex]f(z)=\frac{1}{(z+i)^2(z-i)^2}[/itex]

a past exam question asked me to find and classify the residues of this.
i had to factorise it into this form and then i just said there was a double pole at [itex]z=+i,z=-i[/itex]

now for 5 marks, this doesn't seem like very much work.

is it possible to perform a laurent expansion and then show explicitly that they are poles of order 2 rather than just saying "the power of the brackets is 2 and so it must be of order 2"?
 
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  • #2
If you want to show the explicit expansions you'll need to do two of them. One around the pole at z=i and other around the pole at z=(-i).
 
  • #3
I can't fathom why you would want to compute the Laurent series to find the singularities; just use what you know about poles and arithmetic. Making the problem harder simply for the sake of making it harder is not what mathematics is about...



I note, however, that the question you stated is to find the residues, and you haven't done anything on that...
 
  • #4
so you mean do them seperately?

also how would i expand these? using [itex](1+z)^n=1+nz+\frac{n(n-1)}{2!}z^2+...[/itex]? (what the heck is this expansion called anyway - it's not binomial is it?)
 
  • #5
yeah I've actually done the entire question. I am just wondering if i need to say more about the classification of the poles to get all the marks in the exam or is what i put in post 1 enough?
 
  • #6
Yes, it's enough. E.g. 1/(z-i)^2 is a double pole around z=i and 1/(z+i)^2 is analytic in the neighborhood of z=i. A function doesn't have a single laurent series. It has a different laurent series around every point z=a in the complex plane.
 

1. What is a Laurent expansion?

A Laurent expansion is a representation of a complex function as a sum of a finite number of terms, with each term containing a power of the variable z. It is used to extend the concept of a Taylor series to functions that have both positive and negative powers of z.

2. How do you find the Laurent expansion of a function?

To find the Laurent expansion of a function, you first need to determine the singularities of the function. Then, you can use the formula for Laurent series to calculate the coefficients of the expansion. The expansion will have two parts - a principal part with negative powers of z and a regular part with positive powers of z.

3. What is the difference between a pole and a removable singularity?

A pole is a type of singularity of a function where the function approaches infinity as z approaches a certain value. A removable singularity, on the other hand, is a point where the function is undefined, but can be made continuous by assigning a value to that point. Poles are classified as essential singularities, while removable singularities are not.

4. How do you classify residues in a Laurent expansion?

The residue of a function at a pole is the coefficient of the term with a negative power of z. It can be calculated by finding the limit of the function as z approaches the pole. Residues can be classified as simple, double, triple, etc. depending on the order of the pole.

5. What is the significance of finding residues in a Laurent expansion?

Residues can be used to evaluate complex integrals, which can be helpful in evaluating real integrals that are difficult to solve. They can also provide important information about the behavior of a function at its singularities and help in understanding the properties of the function.

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