Convolution of $e^{-|x|} \ast e^{-|x|}$ I have troubles when I calculate convolution of $e^{-|x|} \ast e^{-|x|}$. 
$$(f\ast g)(x)=\int_{\mathbb{R}}e^{-|y|}e^{-|x-y|}dy$$
I believe I am not defining well boundaries, since final results from Mathlab are:
Case 1: $e^{-x}(x+1)$ for $x>0$;
Case 2: $-e^{x}(x-1)$ for $x \geq 0$
My boundaries for case 1: $$\int_{0}^{x}e^{-x}e^{y-x}dy=e^{-x}(e^{-x}+1)$$
 A: If you want the integral over all of $\Bbb R$ as you say, and as you should to get the convolution, you and Matlab are both wrong. I suspect you entered incorrect code into Matlab. Why would you say the integral over $\Bbb R$ was the same as the integral from $1$ to $x$?
Oh: Also what you wrote for the convolution is wrong. The convolution is$$\int_{-\infty}^\infty e^{-|y|}e^{-|x-y|}dx.$$
You need to determine four sets; the set of $y$ such that (i) $y>0, x-y>0$, (ii) $y<0, x-y>0$, (iii) $y>0,x-y<0$, (iv)$y<0,x-y<0$. You have to figure out the boundaries for each set. On each set you have $|y|=y$ or $|y|=-y$ and $|x-y|=x-y$ or $|x-y|=y-x$. Find all four integrals and add.
(There's a much simpler way using the Fourier transform.)
A: The Fourier transform of $e^{-|x|}$ in $\mathbb{R}^n$ is $c_n\,\frac{1}{(1+|\xi|^2)^{\frac{n+1}{2}}}$, where $c_n$ is some real constant. 
So the Fourier transform of $e^{-|x|} * e^{-|x|}$ becomes $c^2_n\, \frac{1}{(1+|\xi|^2)^{n+1}}$. 
The inverse Fourier transform of $\frac{1}{(1+ 4\pi^2|\xi|^2)^{n+1}}$ can be found in Stein in page number 132, Proposition 2. It says
$$\frac{1}{(4\pi)^{\alpha/2}}\frac{1}{\Gamma(\alpha/2)}\,\int_0^\infty e^{-\pi |x|^2/\delta}\, e^{-\delta/4\pi}\, \delta^{(-n-2+\alpha)/2}\, d\delta = \mathcal{F}^{-1}\left(\frac{1}{(1+ 4\pi^2|\xi|^2)^{\alpha}}\right)$$ for any $\alpha>0$. 
