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

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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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