Calculating the residue of a complex function

[AlBell]

Homework Statement

calculate the residue of the pole at z=i of the function

f(z)=(1+z^2)^-3

State the order of the pole

Homework Equations

I know the residue theorem and also the laurent series expansion but I'm having trouble applying these

The Attempt at a Solution

I think the pole is order 3 but I'm having trouble either expanding the function in order to find the a(-1) term or implementing the residue theorem and would appreciate some help! Thanks

 

[micromass]

Do you know that the residue of a function f at the point a can be calculated by

$$\lim_{z\rightarrow a} (z-a)f(z)$$

 

[AlBell]

I thought it was

?

 

[micromass]

I thought it was

?

Yes, of course, sorry. My formula is for a simple pole.
So, can you use your formula to find the residue?

 

[AlBell]

Yes but I am having trouble actually implementing the formula and wondered if anyone could do an example to help me understand it?

 

[micromass]

Well, first you'll need to determine the order of the pole.

Your guess is that the order of the pole is 3. So you'll need to look at

$\lim_{z\rightarrow a} (z-a)^3f(z)$.

This limit should be nonzero for the order of the pole to be 3. If it is zero, then the pole is of a lower order. If the limit doesn't exist, then the pole is of higher order.

So, can you calculate that limit?

Hint: factor the denominator.

 

[AlBell]

Ah I forgot about factorising, that makes it so much simpler! I get 3/8 I think!

 

[Robert1986]

VERY close. You probably got something like 12*(2i)^(-5), right? Just compute this again. The 3/8 is right, but you are missing two factors.