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Calculation residues at z=1 of order 4

  • Thread starter Kowsi Ram
  • Start date
  • #1
6
0

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)}
 

Answers and Replies

  • #2
938
9

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?).
 
  • #3
6
0
Thanks for your reply. Please can you solve it and show me, sir.
 
  • #4
938
9
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.
 
  • #5
6
0
You are correct. I will try this method.
 

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