Computing an integral -- any method

[Karthiksrao]

Hi,

I have been trying to find an integral $\int_{-\infty}^{+\infty} \frac{ e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2} dx$.

I initially posted this question in the complex analysis forum since I felt it might be done using contour integration. However now I realize it might not be the best way to go about as the integral over the curve in complex space does not converge to zero as the radius of the curve tends to infinity.

Any suggestions on how I can get this integral done?

Thanks!

 

[phyzguy]

I suggest you do it numerically. What level of accuracy do you need? Wolfram Alpha says it is about 0.47569. Is that good enough?

 

[Karthiksrao]

aah, sorry. I do need an analytical result.

 

[phyzguy]

aah, sorry. I do need an analytical result.

I'm curious why. Is it a homework problem, then? It's a definite integral, so the answer is just a number. If you need the number for subsequent calculations, just call it β and plug in the numerical result when you get to the end.

 

[Karthiksrao]

I'm curious why. Is it a homework problem, then? It's a definite integral, so the answer is just a number. If you need the number for subsequent calculations, just call it β and plug in the numerical result when you get to the end.

Nope its not a homework problem. I do research in near-field electromagnetic physics. The original integral has several constants which will later be used for other results. I simplified the original integral to this form. If I solve this problem I will be able to solve the original integral in terms of those constants.