# Finding Laplace inverse transformation of a product series [closed]

compute the inverse Laplace transformation of the following equation.

\begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. \end{align*}

What I managed to get is this \begin{align*} Y^{-1}(\frac{1}{(s+a)^m})&=\frac{t^{m-1}e^{-at}}{{(m-1)!}} \end{align*}

How do I find the Laplace inverse of the series though? Someone suggested me using partial fraction decomposition and gave me this solution, \begin{align*}f(s)=\sum_{1\le k\le m\atop 1\le j\le L} {A_{jk}\over(s+a_j)^k}\end{align*} But failed to tell how did we achieve it.

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## closed as off-topic by Pietro Majer, j.c., Stefan Kohl, Carlo Beenakker, Jan-Christoph Schlage-PuchtaSep 15 at 18:02

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