What are the orders of the poles and the residue for sin(1/z)/cos(z)?

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Homework Help Overview

The discussion revolves around identifying the orders of poles and calculating the residue for the function sin(1/z)/cos(z). The subject area pertains to complex analysis, specifically the calculus of residues.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to identify poles and residues, considering the possibility of a simple pole and the use of Laurent series or limiting cases. They question whether the presence of sin(1/z) affects the classification of the pole at nπ - π/2.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of the problem. Some guidance has been offered regarding the identification of poles, but no consensus has been reached on the implications of sin(1/z) on the pole classification.

Contextual Notes

There is a mention of uncertainty regarding the classification of poles due to the behavior of sin(1/z) and the lack of specific references to textbooks or chapters on the topic.

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Homework Statement


Hello guys, I need to find the orders of each pole as well as the residue of the function sin(1/z)/cos(z).

Homework Equations


I imagine that this is a simple pole so I will either find the Laurent series and get the coefficient of (z-z_0)^{-1} or use the simpler limiting case.

The Attempt at a Solution


So far, I think that there is clearly a pole at n\pi-\frac{\pi}{2} due to the z in the cosine term, although I'm not sure whether it's considered a pole when the value of z causes the term sin(1/z) to go to zero. Any help here and further direction on calculating the residue from there would be awesome. Thank you very much.
 
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What is the name of this chapter ?
 
Theengr7 said:
What is the name of this chapter ?

Calculus of Residues? No particular book.
 
Alright. I have not heard about it before.
 

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