# Complex analysis - an integral with branch cuts

1. Sep 12, 2014

### Loro

1. The problem statement, all variables and given/known data

Hi, I need to calculate the following integral:

$\int_{-\infty}^{+\infty}dx \frac{(\pi+\sqrt{x^2+m^2})^2(1+\cos x)}{(x^2-\pi^2)^2\sqrt{x^2+m^2}}$

3. The attempt at a solution

I tried complexifying it:

$\oint dz \frac{(\pi+\sqrt{z^2+m^2})^2(1+e^{iz})}{(z^2-\pi^2)^2\sqrt{z^2+m^2}}$

And having this over the following contour (sorry for the quality of the image):

https://www.dropbox.com/s/t7ioou1kjs3y7ej/paint.jpg?dl=0

The red dots are the poles at: $\pm\pi$ and $\pm im$.

But is it a valid contour in this case, or should I pay extra attention to the branch cuts of the sqrt in the denominator? If so, how do I choose the contour properly?

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2. Sep 12, 2014

### BiGyElLoWhAt

Well let me start off by saying that improper integrals are by no means my specialty, but I'll try to offer some assistance if that's alright.

I don't know if I would "complexify " the integral like that, as I'm not sure that really simplifies it.

What I am noticing however, is that you have similar terms in the numerator and denominator, and that makes me think expanding your numerator and splitting it up into 2 (or more) fractions might be helpful, as you might get some useful cancellations.

If that didn't work for me, I might try rewriting this whole thing (or maybe the new split up fraction) by splitting the denominator via partial fractions.

Have you tried any of these things? If so I'd be curious to see what it looked like.

3. Sep 12, 2014

\begin{align*} \dfrac{(\pi + \sqrt{x^2 + m^2})^2 (1 + \cos(x))}{(x^2 - \pi^2)^2 \sqrt{x^2+m^2}} & = % \dfrac{(\pi^2 + 2 \pi \sqrt{x^2 + m^2} + (x^2 + m^2))(1+\cos(x))} {% (x^2-\pi^2)^2 \sqrt{x^2 + m^2}} \\ & = T1 + T2 + T3 \text{ where}\\ T1 & = \dfrac{\pi^2 (1+\cos(x))}{(x^2-\pi^2)^2 \sqrt{x^2 + m^2}} \\ T2 & = \dfrac{2 \pi(1+\cos(x))}{(x^2-\pi^2)^2} \\ T3 & = \dfrac{\sqrt{x^2 + m^2}(1+\cos(x))}{(x^2-\pi^2)^2} \end{align*}