The probability density function (PDF) of Gumbel distribution is given as:
$$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x-\mu}{\beta }\right)+\frac{x-\mu}{\beta }\right)\right)}{\beta },$$
where $\beta>0$. I would like to assume the translation $\mu$ to be $0$, which gives:
$$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x}{\beta }\right)+\frac{x}{\beta }\right)\right)}{\beta }.$$
The Mellin transform of the PDF is therefore:
$$\int_0^{\infty } \frac{x^{s-1} \exp \left(-\left(\exp \left(-\frac{x}{\beta }\right)+\frac{x}{\beta }\right)\right)}{\beta } \ dx,$$ where $s>0$.
To solve this, I input the expression into Mathematica but no luck. Is there any other way to solve this?