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

[saybrook1]

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.

 

[Theengr7]

What is the name of this chapter ?

 

[saybrook1]

What is the name of this chapter ?

Calculus of Residues? No particular book.

 

[Theengr7]

Alright. I have not heard about it before.