# Laurent Expansion

Say you have $f(z)=\frac{1}{(z+i)^2(z-i)^2}$

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 $z=+i,z=-i$

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"?

Dick
Homework Helper
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).

Hurkyl
Staff Emeritus
Gold Member
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....

so you mean do them seperately?

also how would i expand these? using $(1+z)^n=1+nz+\frac{n(n-1)}{2!}z^2+...$? (what the heck is this expansion called anyway - it's not binomial is it?)

yeah ive actually done the entire question. im 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?

Dick