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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.

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    I think the summation does not converge because it is highly oscillatory...2011-06-10

1 Answers 1

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Using the substitution $t=\exp(x)$, your integral can be brought into the form $\mathcal{I}= \int_1^\infty \frac{e^{-\beta t}}{t^{\mu+1}} dt=E_{\mu+1}(\beta)$ with $E_n(z)$ the exponential integral. If $\beta$ and $\mu$ are large, we can use the expansion $E_n(z)= \frac{e^{-z}}{z+n} \left[ 1 + \frac{n}{(z+n)^2} + \frac{n(n-2z)}{(z+n)^4} + \frac{n (6z^2 -8 n z + n^2)}{(z+n)^6} + \dots \right].$ If $|\beta| = |\mu| = 10^3$, each term is $10^{-3}$ smaller than the proceeding term so that I would expect that you need only the first few terms (maybe only the first).

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    I think it will hold also for complex values in some sector of the complex plane...2011-06-11