We can start as follows, by transforming it into a generalized hypergeometric function:
, since, from the series expansion of the generalized hypergeometric function, , where is the Pochhammer symbol .
Now the integrand function does not appear to be convergent numerically, except for where it becomes the Gaussian integral, and the case of where it becomes a Bessel function. For and , the integrand takes values of (serious). Beyond that the computer starts to produce smoke. Yet it eventually converges as there is a closed form solution. It is like saying that it works in theory but not in practice!
For, it turns out, under the restriction that , we can use the following result:
Allora, we can substitute , and with , given that ,
So either the integrand eventually converges, or I am doing something wrong, or both. Perhaps neither.