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I would like to compute the following integral:

$$\int_{-\infty}^{\infty}\frac{dx}{2\pi} \frac{e^{-2d\sqrt{-x^2+\alpha^2+i\epsilon}}}{(-x^2+\alpha^2+i\epsilon)^2}$$

where $d, \alpha, \epsilon > 0$ and $d \alpha >>1$ and $\epsilon << 1$ (I will keep to first order in $\epsilon$ throughout).

My approach has been to analytically continue the integral and perform a contour integration. This integral will have branch points at $z = \pm (\alpha + i\epsilon)$. I have so far assumed that because $z = \pm (\alpha + i\epsilon)$ are branch points, they are NOT also poles of order $2$ as they otherwise would be. I have tried taking a key hole contour in the lower half plane around the branch point at $z=-\alpha-i\epsilon$ but do not get anywhere with this.

Is my attempt so far valid? Can anyone help me in computing this?

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  • $\begingroup$ When say "compute", do you really mean "find the asymptotic", given $d \alpha >>1$ and $\epsilon << 1$? $\endgroup$ Commented Oct 3 at 12:31
  • $\begingroup$ Yes, ultimately I’m am interested in the leading order behaviour in these limits. If it is possible, it would also be nice to have an analytical expression for the integral. $\endgroup$ Commented Oct 6 at 6:04
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    $\begingroup$ I think the asymptotic will very much depend on how big or small $d\,\epsilon$ is. In this sense, the condition that $\epsilon$ is small may not be enough. $\endgroup$ Commented Oct 6 at 19:39
  • $\begingroup$ If I also stipulate that $\epsilon << d$ and that $\epsilon d << 1$, does this help? $\endgroup$ Commented Oct 10 at 11:55
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    $\begingroup$ You may consider asking this at math.stackexchange.com, where this type of question is often answered. This is a 'complex' situation, where you have a branch point coinciding with a pole, both of which must be considered. This is best handled with the methods of uniform asymptotics. I would also like to see this answered, as I have been attempting a similar problem. $\endgroup$ Commented Oct 10 at 16:44

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