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While solving different contour integrals, I stumbled upon quite a few line integrals with pole(s) on contour. I've always solved them the same way, using the rule that for such lines the integrals equal

[tex]\int\limits_\gamma f(z)dz=\pi i\sum\limits_n Res(f,p_n),[/tex]

where [itex]p_n[/itex] are the poles, that is half the contour integral around all the poles.

In my homework (I am a PhD student), I encountered the following integral:

[tex]\int\limits_{-\infty}^{\infty}\frac{\cos(bx)-\cos(ax)}{x^2}dx.[/tex]

I am using the following contour to calculate it:

It is a very easy integral to calculate if I just use the mentioned rule. First, I write it down as follows:

[tex]PV\int\limits_{-\infty}^{\infty}\frac{e^{ibx}-e^{iax}}{x^2}dx,[/tex]

where I am using the fact that [itex]\sin(x)[/itex] is an odd function, and the integral is taken as a principal value. It is easy to prove that in the limit of [itex]R\to \infty[/itex] the semi-circle integral vanishes and so we can say that our integral is simply [itex]\pi i[/itex] times the residue at 0. The result matches the provided answer in the book.

The problem, however, is that on the lectures, the above mentioned rule has never been introduced (neither has VP, truth said, but I think it does not need an introduction in this context). As such, I feel that I need to provide, at least, some informal proof of it. Our professor doesn't require us to do all formal math, but she does expect us to, at least, verbally explain what we did and why.

No matter how hard I tried, I haven't managed to come up with any reasonable "intuitive" explanation. Extensive search in the Internet hasn't yielded anything either. The only theorem I found that was any close to what I need is Plemelj theorem, but I don't think I can prove it myself, and also the proof will probably be too complex for such a simple problem. Also I know that there is some theorem I encountered a couple of years ago that said something about functions defined at points of non-continuity as

[tex]f(x_0)=\frac{1}{2}(\lim\limits_{x\to x_0-0}f(x)+\lim\limits_{x\to x_0+0}f(x)),[/tex]

which could be used in this case as I take the limit of the bottom line of the contour to the x axis from above and from below it. But I haven't found the exact formulation of it.

Could someone please link something relevant to me that would help me justify this formal rule of integration? Ideally I would like to justify it universally, for all integrals, although a justification for this particular integral would also be helpful. Thanks!

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# Contour integral with a pole on contour - justification?

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