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Is there any way to solve this integral?

$$\int_{-\infty}^{\infty}\frac{e^{qiy -K(\sqrt{\lambda -a-iy}-\sqrt{\lambda})}}{a+iy} dy$$ where $K,\lambda, a$ and $q$ are real numbers and $K>0$, $a>0$, $\lambda > 0$ and $q<0$

I have tried the standard contour approaches, but the branch cut makes it complicated on the lower half plane, and the integrand grows unbounded on the upper half plane.

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No assumptions on $a$ and $\lambda$? If we use the branch cut of the square root along the negative real axis, I suppose $\lambda-a$ had better have a positive real part. (And I find it easier to think of this as an integral along the imaginary axis wrt $z=iy$.) – Harald Hanche-Olsen Mar 28 '12 at 14:33
You are right, otherwise it will be even nastier. I forgot to write the assumptions for $\lambda$ and $\alpha$, I added them now. – Jo.Bent Mar 28 '12 at 15:02
Let the quantity underneath the radical sign equal $x^2$. Then your integral becomes a combination between the Gaussian and exponential integral. – Lucian Dec 11 '13 at 3:05

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