# Definite integral of Gumbel functions

I need to calculate the following definite integral of Gumbel functions:

$$\int_{-\infty}^{+\infty}e^{\frac{x-\alpha}{\beta}}e^{-e^{\frac{x-\alpha}{\beta}}}e^{-e^{-\frac{x-\gamma}{\delta}}}dx,$$

given real parameters $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$. In particular $$\beta,\delta > 0$$. I tried to apply the following change of variable:

$$z=e^{\frac{x-\alpha}{\beta}},$$

so that $$dz=\frac{1}{\beta}zdx$$ and $$e^{-\frac{x-\gamma}{\delta}}=az^{b}$$, where $$a=e^{\frac{\gamma-\alpha}{\delta}}$$ and $$b=-\frac{\beta}{\delta}$$. In this way, the integral can be written as follows:

$$\beta\int_{0}^{+\infty}e^{-z}e^{-az^{b}}dz,$$

where $$b<0$$. I've tried to calculate this integral by expanding one of the exponential functions in a Taylor series, for example:

$$\beta\sum_{n=0}^{+\infty}\frac{\left(-a\right)^{n}}{n!}\int_{0}^{+\infty}z^{nb}e^{-z}dz.$$

Each integral in the series looks like a Gamma function, but unfortunately they diverge, since $$b<0$$. According to WolframAlpha, for some special values of the parameter $$b$$, this integral equals a very complicated Meijer G-function. Is there a closed-form formula or a convergent series expansion for this integral (or for the original one with the Gumbel functions), given arbitrary values of the parameters $$a$$, $$b$$? Thanks in advance for any help you can provide.

• I'm almost certain there's no closed form in general – Yuriy S Oct 30 '18 at 16:59
• – Yuriy S Oct 30 '18 at 17:01
• By following a suggestion in one of the links that you provided, maybe it could be possible to express the integrand $e^{-\left(z+az^{b}\right)}$ as a (product of) Meijer G-function(s) and use the general integration formula shown here docs.sympy.org/latest/modules/integrals/g-functions.html – user2983638 Oct 31 '18 at 9:11

The M-Wright function can be written as $$W_{\lambda,\mu} = \sum_{n=0}^\infty \frac{z^n}{n!\,\Gamma(\lambda\,n+\mu)}, \quad \lambda>-1 .$$ It is clear the last expression can be put in this form, with the use of the Gauss formula for the $$\Gamma$$-function. I'm not very familiar with this special function, but it has been around from the 1930s and has a searchable literature on the internet.
• According to this link math.stackexchange.com/questions/204181/…, when $b>0$ the integral corresponds to a Fox-Wright function, but unfortunately $b<0$ in our case. – user2983638 Oct 31 '18 at 9:01