Knowing that the laplace transform of a Gamma distribution is given by:
$$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$
and that for Z = X + Y "Sum of two independent Gamma distribution random variables" $$F_z(s) = (\frac{\beta_x}{s + \beta_x})^a (\frac{\beta_y}{s + \beta_y})^b$$
I am trying to evaluate W= X - Y in the S domain for both X and Y are positive.
1- Is it correct to write: $$F_w(s) = (\frac{\beta_x}{s + \beta_x})^a (-1)( \frac{-\beta_y}{s - \beta_y})^b$$
2- If I am interested in the Inverse Laplace (I know it is not easy) how can I get to the two domains namely "$w>0$" and "$w<0$" from one S function. I don't want to get the actual expression of $f_z(z)$ just how is it transformed into two domains.
3- What if $s=\beta_y$ what will be the value or the meaning of $F_w(s)$
The distribution of $W=X-Y$ for independent $X\sim\operatorname{Gamma}(\alpha_1,\beta_1)$ and $Y\sim\operatorname{Gamma}(\alpha_2,\beta_2)$ is known to be \begin{equation} \label{eq:GamDiff_general} f_W(y) = \begin{cases} \frac{C_W}{\Gamma(\alpha_2)}\,e^{\beta_2w}\,U(1-\alpha_2,2-\alpha_o,-\beta_ow) , &w \leq 0,\\ \frac{C_W}{\Gamma(\alpha_1)}\,e^{-\beta_1w}\,U(1-\alpha_1,2-\alpha_o,\beta_ow), &w > 0, \end{cases} \end{equation} where $\alpha_o=\alpha_1+\alpha_2$, $\beta_o=\beta_1+\beta_2$, and $C_W=\beta_1^{\alpha_1}\beta_2^{\alpha_2}\beta_o^{1-\alpha_o}$. To get the laplace transform of this pdf you can use DLMF 13.10.7 with $\nu=1$. The solution will be the sum of two functions of $s$.