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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 $z=iy$.)2012-03-28
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    You are right, otherwise it will be even nastier. I forgot to write the assumptions for $\lambda$ and $\alpha$, I added them now.2012-03-28
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    Let the quantity underneath the radical sign equal $x^2$. Then your integral becomes a combination between the [Gaussian](http://en.wikipedia.org/wiki/Gaussian_integral) and [exponential integral](http://en.wikipedia.org/wiki/Exponential_integral).2013-12-11

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