How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution? 
How to show that, for two random variables $X,Y\sim\text{Gumbel}[0,1]$, $X-Y\sim\text{Logistic}[0,1]$?

I tried to use the convolution formula $$\int_{-\infty}^{\infty}f_X(w)f_{-Y}(z-w)dw$$ I obtain $$\int_{-\infty}^{\infty} e^{-e^{-w}-e^{z-w}+z-2w}~dw$$ However, I cannot find the antiderivative of this expression.
Now I have two questions:


*

*Am I taking the right approach to show this?

*Can you give me a hint how to solve the integral or recommend any literature that might help?

 A: The change of variables $u=\mathrm e^{-w}$ transforms this into a neat gamma integral, to which one can apply the change of variable $v=(\mathrm e^z-1)u$ to conclude. 
Note that the transformation $w\to\mathrm e^{-w}$ is ubiquitous in quite a few Gumbel related manipulations.
A: This can also be done quite simply using characteristic functions.
The characteristic function for the Gumbel($\mu$,1) is
$\Phi(t)=e^{i\mu t}\Gamma(1-it)$.
For independent variables X and Y, their linear combination has characteristic function:
\begin{align}
\Phi_{X-Y}(t)&=\Phi_{X}(t)\Phi_{Y}(-t)\\
&=e^{i\mu_X t}\Gamma(1-it)e^{-i\mu_Y t}\Gamma(1+it)\\
&=e^{(\mu_X-\mu_Y) it}\Gamma(1-it)it\Gamma(it)
\end{align}
where the last line uses the fact that the Gamma function satisfies the functional equation $\Gamma(1+z)=z\Gamma(z)$.
We then use Euler's Reflection Formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ to get
\begin{align}
\Phi_{X-Y}(t)&=e^{(\mu_X-\mu_Y) it} i t\frac{\pi}{\sin(\pi it)}\\
&=e^{(\mu_X-\mu_Y) it} \frac{\pi t}{-i \sin(\pi it)}\\
\end{align} 
which is the characteristic function for the logistic distribution.
