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March 15th, 2012, 03:03 AM  #1 
Newbie Joined: Jan 2011 Posts: 20 Thanks: 0  Mistake with a simple residue problem!
(First I will forget the that is required when calculating every residue as it is not important here) Calculate the residue at the simple pole for the function If I use "The Residue at a Pole" lemma which states: Let be a pole of order of so that , for some g, analytic at with , then For my function, I have simple poles (which is why I post a simpler an nongeneral version of the lemma), but when I apply this Lemma to find residue at my pole, I have: This gives me: But this is incorrect! The correct answer is Where is this 1/2 coming from?? And similarly, I face another problem: Find the residue at the point for the function At 1/3 I have a pole of order 2, so I rewrite , differentiate (because I do not have a simple pole, it is of order 2 => differentiate once) and insert in , which gives me But this is also incorrect, as the correct value is , which is ... Where are these 1/2 and 1/9 terms coming in?? 
March 15th, 2012, 07:19 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Mistake with a simple residue problem! [color=#000000]Ok you confused me! You have , for the other one you have a second order pole at z=3. Use the formula , in our case n=2 and for the pole at write f in the following form and follow the above steps .[/color] 

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