# Is this Fourier integral well-known?

The following integral is a special case of one that arises in an economics problem:

$I(u_{1}, u_{2}) := \displaystyle \int_{z_{1}=-\infty}^{\infty} \int_{z_{2}=-\infty}^{\infty} \frac{ \displaystyle e^{iu_{1}z_{1}+iu_{2}z_{2}} \Gamma(a + iz_{1} + iz_{2})\Gamma (a-iz_{1}) \Gamma(a-iz_{2}) }{ \displaystyle \lambda - \left\{ (a-1+iz_{1}+iz_{2})^{2} + (a-iz_{1})^{2} + (a-iz_{2})^{2} \right\} } \, dz_{1} \, dz_{2}$

$u_{1}$ and $u_{2}$ are real numbers. $\lambda$ and $a>0$ are also real numbers, to be thought of as parameters, which obey the condition $\lambda - \left\{ (a-1)^{2} + a^{2} + a^{2} \right\} >0$.

Is this integral well known? Does it arise in other contexts? Can it be simplified in any way?

The one-dimensional analogue of this integral reduces to hypergeometric functions, if that helps.

My second question is:

What can be said about the qualitative behaviour of the integral, for example in the limit as $u_{1} \to \infty$, or in the limit as $u_{1}, u_{2} \to \infty$ with $u_{1}-u_{2}=K$, some constant?

I expect more interesting behaviour in the regime in which $\lambda$ is small, so $\lambda \approx (a-1)^{2} + a^{2} + a^{2}$.

EDIT: I would also be interested to be told that there is no reason to expect this to be tractable. The more symmetric integral with the -1 deleted from the denominator is also of some interest, in the unlikely event that that makes it easier to analyze...

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