Residue Calculus: Evaluating Poles and Contours

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    Calculus Residue
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Homework Help Overview

The discussion revolves around evaluating poles and contours in the context of residue calculus. Participants are examining the locations of poles and their relation to specified contours in complex analysis.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the identification of poles and their orders, questioning whether certain poles lie within specified contours. There is an emphasis on visualizing the problem through graphing.

Discussion Status

Some participants express uncertainty about the inclusion of poles within contours, with guidance suggesting the use of graphical methods. There appears to be a productive exchange regarding the identification of poles in different scenarios, though no consensus is reached on all points.

Contextual Notes

There is mention of a tutor's advice to draw graphs to aid in understanding the problem, indicating a reliance on visual aids to clarify the positions of poles relative to contours.

eeuler
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Homework Statement


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The Attempt at a Solution


So there are poles at: z=[tex]\pm2[/tex] and at z= -1 of order 4. Right?

My query is, when evaluating these poles (using the residue theorem), is it right that for (i) Z = 1/2, no residues lie in that contour?
for (iii), do all residues lie in the contour?
 
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Right about the poles, but for the rest, are you just guessing? Why don't you draw a graph? No, I don't think the triangle in iii) includes all poles.
 
^^I wasn't sure how to determine whether a pole is inside a specified contour but my tutor suggested drawing a graph, which i have been doing, in which case shouldn't (i) have no poles in the contour? As for (iii) then, would Z=+2, and Z=-1 lie in the contour?
 
Your tutor is wise. Yes, i) contains no poles and iii) contains z=(-1) and z=2.
 
^^Thanks :)
 

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