Evaluating Integrals: Need Help Factorising Denominator

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Homework Help Overview

The discussion revolves around evaluating an integral that involves complex variables and requires factorization of the denominator. Participants are exploring methods to simplify the expression and identify poles.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to factorize the numerator of a fraction derived from the integral but expresses uncertainty about the process. Some participants suggest multiplying by a variable and solving a second-order equation, while others provide complex integral expressions for consideration.

Discussion Status

The discussion is ongoing, with participants offering different approaches to the problem. There is an indication of helpful guidance being provided, but no consensus has been reached regarding the best method to proceed.

Contextual Notes

Participants are navigating through complex variable integration and the implications of poles in the context of the integral. The original poster has indicated a need for clarification on the factorization process and the identification of poles.

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Homework Statement



Evaluate the integral
qywvtv.png


Homework Equations


I can substitute
s4q5pc.png
and thus end up with
2j33dix.png


The Attempt at a Solution


I then expand the denominator out and end up with 1/
kd33ap.png

However I then assume I need to factorise the top line of that fraction as this will be the denominator in my integral, so I want to factorise this and find where it equals zero and thus where the poles are. However I'm unsure how to do this and need help! Thanks
 
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Just multiply upstairs and downstairs by z, carry z^2 in downstair inside the brackets, and solve the 2nd order equation.
 
sorry I don't really understand what you are saying!
 
[tex]I = -i \oint_{|z|=1} \frac{dz}{z(5-3(\frac{z-z^{-1}}{2i}))^2} = -i \oint_{|z|=1} \frac{ z dz}{\left[5z-3(\frac{z^2-1}{2i})\right]^2}[/tex]
and then solve
[tex]5z-3(\frac{z^2-1}{2i}) = 0[/tex]
(and remember how many of each pole there are!)
 
thanks! I'll give it a go
 

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