EmberCult
09142018
08:57 AM ET (US)

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^{m1}e^{at}}{{(m1)!}} \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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