I'm faced with the integral
$\mathcal{I} = \int\nolimits_0^\infty \mathrm d x \; e^{-\beta \, e^x - \mu x} \;,\quad \Re(\beta) > 0 \;.$
The solution can be looked up. It reads
$\mathcal{I} = \beta^\mu \, \Gamma(-\mu,\beta) = \beta^\mu \left( \Gamma(-\mu) + \frac{\beta^{-\mu} e^{-\beta}}{\mu} M(1, 1 - \mu, \beta) \right) \;,$
where $\Gamma(-\mu,\beta)$ is the incomplete gamma function and $M(1, 1 - \mu, \beta)$ is Kummer's confluent hypergeometric function.
My question is, is there any way to scale the coefficients $\beta$ and $\mu$ within $\mathcal{I}$ to drag out a common factor? The problem is, that in my case the imaginary parts of $\beta$ and $\mu$ are about $10^3$, which makes the numerical evaluation of the above expression quite complicated. For one thing,
$\beta^\mu = e^{\mu \ln \beta} \sim e^{10^3} \;,$
which cannot be calculated using standard floating point numerics. In addition the direct summation of $M(1, 1 - \mu, \beta)$ does not seem to converge, which is probably related to large round-off errors again caused by the fact, that $\beta$ and $\mu$ are large.
I thought there must be some way to perform a coordinate transform within $\mathcal{I}$, but up to now I couldn't come up with something that doesn't destroy the general form of the integral.