# Inverse Laplace transform of the Meijer G-function

$$\mathcal M_Y(s)=\frac{1/\sqrt\pi}{\prod_{i=1}^N\Gamma(m_i)}G_{2,N}^{N,2}\left[\frac4{s^2}\prod_{i=1}^N\left(\frac{m_i}{\Omega_i}\right)\Bigg|_{m_1,m_2,\dots,m_N}^{1/2,1}\right]\tag{3}$$ $$f_Y(y)=\frac2{y\prod_{i=1}^N\Gamma(m_i)}G_{0,N}^{N,0}\left[y^2\prod_{i=1}^N\left(\frac{m_i}{\Omega_i}\right)\Bigg|_{m_1,m_2,\dots,m_N}\right]\tag{4}$$ (Pic 1, Pic 2)

Can any one tell me how they got equation (4) out of (3)? Attached is the equations. They say that they did the inverse laplace trasnform to get it, but i don'y know how! if any one used Mathematica to solve it, that would be more than enough. where equation(3) is the moment generating function and equation (4) is the pdf.

• @Parcly Taxel: you have removed equation 3 and put (4) again instead of it !! and now i can't edit it. will you please return it back? – user42138 Aug 26 '16 at 10:34
• Have added back equations. – Parcly Taxel Aug 26 '16 at 10:35
• @ParclyTaxel: okay, thanks – user42138 Aug 26 '16 at 10:36