Have an Integral that I can't seem to solve

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Discussion Overview

The discussion revolves around a challenging integral encountered in a quantum mechanics class. Participants explore methods for solving the integral, which involves complex analysis concepts, particularly residue theory. The integral is presented with specific bounds and a mathematical expression that some participants find difficult to interpret or solve.

Discussion Character

  • Homework-related
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant presents an integral that they are struggling to solve, seeking assistance from others.
  • Another participant reformulates the integral for clarity, confirming the expression provided by the original poster.
  • A participant mentions that the integral is standard in complex analysis and suggests that knowledge of residue theory is necessary for solving it.
  • One participant expresses surprise that the problem requires knowledge of residue theory, noting that it has not been covered in their previous math classes.
  • Another participant reiterates the importance of residue theory as a standard topic in calculus of complex variables, indicating that it is typically encountered by undergraduate students by their second year.

Areas of Agreement / Disagreement

Participants generally agree that the integral is related to complex analysis and that residue theory is relevant for solving it. However, there is no consensus on the accessibility of the problem given the participants' varying backgrounds in mathematics.

Contextual Notes

Some participants express uncertainty about the integral's solution and the prerequisites for understanding residue theory. There is a noted lack of exposure to complex analysis in the participants' educational backgrounds, which may affect their ability to tackle the problem.

maverick_76
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So I am working on a problem for my quantum mechanics class and I cannot seem to figure out how to solve this integral. I have looked up tables and entered it into Wolfram and it doesn't give me anything that I can work with. The integral is as follows:∫ 2aπ(cos(ak)+1)
(a^2 k^2- π^2)^2

And the bounds are from -∞ to ∞
Any help on this would be greatly appreciated. Sorry for the crappy equation layout, I'm not sure how to use LaTeX.
 
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Is this integral perhaps:

$$\int_{-\infty}^{\infty} \frac{2aπ\; (cos(ak)+1)}{(a^2k^2-π^2)^2}dk$$ ?
 
yes, thak you!
 
It is a pretty standard integral in complex analysis, and it is quite easy if you knoe how to compute integrals using residuies. If you know residues, that is a standard exercise, but if you do not no complex analysis I do not know any "elementary" ways to compute it.
 
Yeah I was told that by the TA for the class that I should look up residue theory, never heard of it until today. Kinda surprised that our professor would put a problem that required that considering every math class I've taken has never touched the subject. Thanks for the input!
 
maverick_76 said:
Yeah I was told that by the TA for the class that I should look up residue theory, never heard of it until today. Kinda surprised that our professor would put a problem that required that considering every math class I've taken has never touched the subject. Thanks for the input!
Residue theory is one of the standard topics covered in calculus of complex variables. It's usually something one encounters by at least second year of undergrad ...
 

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