Have an Integral that I can't seem to solve

[maverick_76]

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.

 

[SteamKing]

Is this integral perhaps:

$$\int_{-\infty}^{\infty} \frac{2aπ\; (cos(ak)+1)}{(a^2k^2-π^2)^2}dk$$ ?

 

[maverick_76]

yes, thak you!

 

[Hawkeye18]

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.

 

[maverick_76]

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!

 

[SteamKing]

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 ...