I need some help calculating the analytically expression of this convolution.
The functions in question are:
1) a gaussian ($g(x)$)
2) a "double tophat function" (in lack of a better name). i.e. $$ f(x)= \begin{cases} a\quad&\text{if } |x|<x_0,\\ b &\text{if } |x|<x_1,\\ 0&\text{otherwise}, \end{cases} $$ where $a>b$ and $x_1>x_0$.
I've tried various analytically tools but none of them has given me a useful answer.
Can someone please help me. I have no interest in the numerical solution. The reason I need it is that I want to analyze the behavior of the convolution for different $a$'s, $b$'s $x_0$'s and $x_1$'s of ($(g*f)(x)$)
As an example:
$$ \int_{x_0/2}^{x_1/2} dx' \: \exp{\left (-\frac{(x-x')^2}{w^2} \right ) = } \frac{1}{2} \sqrt{\pi } w \left(\text{erf}\left(\frac{x-\frac{x_0}{2}}{w}\right)-\text{erf}\left(\frac{x-\frac{x_1}{2}}{w}\right)\right) $$