Inverse Laplace Transforms

In general, finding an inverse Laplace transform in terms of elementary functions is difficult.  For easy problems, we consult a table of Laplace transforms and attempt to algebraically manipulate equations into structures we already find in the Laplace transform side of the table and then use linearity, homogeneity and the fact that we know the Laplace transforms of many elementary functions to calculate the inverse Laplace transform.  There are certainly other methods like the Mellin inverse formula (also commonly referred to as the Bromwich Integral) and we will investigate these in the future. 


Now that we have calculated a number of Laplace transforms of particular functions, we can imagine making a table of all such reasonable functions.  I will abuse notation a little bit here as well.  In general, we use the one-sided Laplace Transform and so the inverse transform often results in functions in terms of the Heaviside function.   I will generally assume we are tacitly considering a domain where t ≥ 0 and if we live with this assumption then we get easier to read equations back.  Sometimes, this will not be possible or useful though.

Let's consider a nice elementary example where we calculate an inverse Laplace transform.


The real utility of this comes in solving differential equations.  We rearrange the transform in terms of F(s) and elementary functions of s on right hand side.  Often, we have to use partial fraction decomposition to assist us here.  Let's look at another example:


I really do want to emphasize that it's very to generate situations where it is hard to find the inverse Laplace transform.  Consider, for instance, ℒ   {cos(s)} is not definable in terms of elementary functions.