Calculation residues at z=1 of order 4

[Kowsi Ram]

Homework Statement

Find the residue of the following function at z=1

Homework Equations

f(z)=z^2/(z-1)^4(z-2)(z-3)

The Attempt at a Solution

lim z→2 z^2/(z-1)^4(z-3)=-4

limz→3 z^2/(z-1)^4(z-2)=9/16

at z=1 of order 4?? using formula 1/m-1! d/dz{(z-a)^m f(z)}

 

[clamtrox]

Homework Statement

Find the residue of the following function at z=1

Homework Equations

f(z)=z^2/(z-1)^4(z-2)(z-3)

The Attempt at a Solution

lim z→2 z^2/(z-1)^4(z-3)=-4

limz→3 z^2/(z-1)^4(z-2)=9/16

at z=1 of order 4?? using formula 1/m-1! d/dz{(z-a)^m f(z)}

Looks good, except that you need to take m-1 derivatives (why?).

 

[Kowsi Ram]

Thanks for your reply. Please can you solve it and show me, sir.

 

[clamtrox]

I think it would be better if you did your homework yourself.

Remember the definition of a residue -- there's an easier way to calculate it in this case, than differentiating the function three times. Expand it into a Laurent series around z=1 by multiplying a couple of geometric series together.

 

[Kowsi Ram]

You are correct. I will try this method.