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

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In summary, the conversation is about finding the orders of each pole and the residue of the function sin(1/z)/cos(z). The approach discussed includes using the Laurent series or the simpler limiting case. The presence of a pole at n\pi-\frac{\pi}{2} is mentioned, but there is uncertainty about its classification. The name of the chapter being discussed is not specified.
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saybrook1
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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 [tex](z-z_0)^{-1}[/tex] or use the simpler limiting case.

The Attempt at a Solution


So far, I think that there is clearly a pole at [tex]n\pi-\frac{\pi}{2}[/tex] 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 ?
 
  • #3
Theengr7 said:
What is the name of this chapter ?

Calculus of Residues? No particular book.
 
  • #4
Alright. I have not heard about it before.
 

1. What is the residue of sin(1/z)/cos(z)?

The residue of a function at a given point is the coefficient of the term with the highest negative power in the Laurent series expansion of the function around that point. In this case, the residue of sin(1/z)/cos(z) at z=0 is equal to 1, as the Laurent series expansion of this function has a term with a coefficient of 1/z.

2. How is the residue of sin(1/z)/cos(z) calculated?

The residue of sin(1/z)/cos(z) at a given point can be calculated using the formula: Res(f,z) = lim[z→z0] [(z-z0)f(z)], where f(z) is the given function and z0 is the point of interest.

3. Can the residue of sin(1/z)/cos(z) be negative?

Yes, the residue of sin(1/z)/cos(z) can be negative. The sign of the residue depends on the order of the pole at the given point. If the pole is of order n, then the residue will have a positive sign if n is odd and a negative sign if n is even.

4. What is the significance of the residue of sin(1/z)/cos(z)?

The residue of a function at a given point is important in the study of complex analysis as it provides information about the behavior of the function around that point. It is used in the evaluation of integrals, calculation of residues at infinity, and in the proof of Cauchy's residue theorem.

5. How does the residue of sin(1/z)/cos(z) relate to the concept of poles?

The residue of a function at a given point is closely related to the concept of a pole. A pole of a function is a point where the function is undefined or infinite. The residue at a pole is the value of the function at that point, which can be calculated using the formula mentioned above. In this case, the function sin(1/z)/cos(z) has a pole of order 1 at z=0, and its residue at that point is equal to 1.

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