Why do poles occur at specific values of q in algebraic function integration?

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Discussion Overview

The discussion revolves around the integration of an algebraic function with respect to a variable q, particularly focusing on the presence of poles at specific values of q, such as q = A, and the implications for using the residue method in the context of finite limits. Participants explore the nature of these poles and their relation to the periodicity of the cosine function within the integral.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that poles occur at specific values of q, particularly where the denominator becomes zero, such as at q = A.
  • Another participant questions the applicability of the residue method given the finite limits of the integral.
  • Concerns are raised about the periodic nature of the cosine function, suggesting that it equals E an infinite number of times, which complicates the identification of poles.
  • There is a suggestion that the integral may be analyzed as part of a closed path in the complex plane, which could affect the use of residue methods.
  • Some participants express uncertainty about their knowledge of complex analysis and the residue theorem, indicating a need for clarification on how to approach the integration.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the integration process and the application of the residue method. There is no consensus on how to handle the infinite occurrence of poles or the best approach to the integral.

Contextual Notes

The discussion highlights limitations related to the periodicity of the cosine function and the implications for pole identification within the context of the integral. There are also unresolved questions about the application of residue theory in this scenario.

Who May Find This Useful

This discussion may be of interest to those studying complex analysis, particularly in relation to integration techniques involving poles and residues, as well as students encountering these concepts in physics applications.

Master J
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Say I am integrating some algebraic function with respect to a variable q. There is also an exp( iq ) in the integral as a factor.

There are poles when q takes a certian value as it is in the denominator as q - A, so pole at q = A.I know the poles are in the upper left plane, and lower right , but I can't see why this is. Could someone explain this ?

In addition, there are finite limits to the integral. Can one still use the residue method on this?

Cheers!

PS

If someone would like to also show me how to integrate it, I would be ever so grateful:

Integral of dq. exp ( iqd) / cos(qd) - E between pi/d and -pi/d
 
Last edited:
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[tex]\int[/tex][tex]^{d/pi}_{-d/pi}[/tex]exp( iqd) / cos(qd) - E dq
 
Master J said:
[tex]\int[/tex][tex]^{d/pi}_{-d/pi}[/tex]exp( iqd) / cos(qd) - E dq

[tex]\int^{d/\pi}_{-d/\pi} \frac{e^{iqd}}{\cos(qd)-E}dq[/tex]

click on the formula and you can see how its tex'd
 
The wonders of Latex! :-) Thanks for putting that straight.


Now that the integral is clear, can anyone share with me HOW one does it??
 
yeah sorry. My residue knowledge is a little rusty, I was paying attention to follow along when someone else answers :p
 
No problem. Hopefully someone here can enlighten me? I haven't done much Complex Analysis, I've just recently encountered it in a course on the application of Green's Functions to physics, so I am at a loss here!
 
I am a little puzzled. Since cosine is periodic, it seems to equal E an infinite number of times, and where this happens should depend on E.

If you want to look at your integral as along an interval on the x axis, and as part of some closed path in the plane, it does not lend itself to the usual method of residues since the poles seemingly keep occurring more and more often as you try to go to infinity. but maybe you can take a vertical rectangle, based on your interval and maybe there will only be a finite number of poles in there.

I would look in some complex book like churchill.
 

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