Contour Integration Homework: Find Poles for Integrals

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Homework Help Overview

The discussion revolves around identifying poles in contour integrals involving complex functions. The original poster presents integrals with specific forms and queries about the existence of additional poles beyond the one identified at -(\gamma+\gamma_{n}).

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the definition of poles and question the validity of the original poster's identification of poles. There is also inquiry into the appropriate integration technique for evaluating the integrals presented.

Discussion Status

The discussion is active, with participants exploring different interpretations of the poles and the integration method. Some guidance has been offered regarding the use of the residue theorem, though there is uncertainty about the expressions resulting from solving the quadratic equations involved.

Contextual Notes

Participants note the complexity of the integrals and the potential for multiple poles, indicating a need for careful analysis of the denominators involved. There is also mention of the challenges in expressing the solutions derived from the quadratic equations.

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Homework Statement


Find the poles for the integrals:
[tex] \int_{c}ds e^{st} \frac{1}{(s + \kappa + \frac{g^{2}}{s+\gamma+\gamma_{n}})} [/tex]

One pole is at [tex]-(\gamma+\gamma_{n})[/tex].

Is there another one?Also
[tex] \int_{c}ds e^{st} \frac{1}{(s + \kappa + \frac{g^{2}}{s+\gamma+\gamma_{n}})} \frac{1}{s+\gamma+\gamma_{n}}[/tex]
Similiarly, one pole is at [tex]-(\gamma+\gamma_{n})[/tex].

Is there another one? I just need to be sure.

Thanks

Homework Equations


The Attempt at a Solution

 
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Poles are the values of the integration variable s where the denominator vanishes, right? I don't think -(gamma+gamma_n) is a pole. In both cases, it looks to me like you are getting a relatively complicated quadratic in s.
 
Can you tell me what technique of integration I should use to evaluate this integral?

[tex] \noindent\(\frac{e^{s t}}{g^2+(s+\text{yn}+\gamma ) (s+\kappa )}\)[/tex]
 
The ordinary residue theorem, I think. You can write that as e^(st)/((s-p1)(s-p2)) where p1 and p2 are the poles you get from solving the quadratic equation. It's straightforward in principle, but I don't know a simple way to write the nasty expressions you get from solving the quadratic. Where are you getting these problems??
 

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