The Inverse Laplace Transform

Part 1: Relation of Laplace and Fourier Transforms

In this module we will use the Residue Theorem of complex analysis to obtain inverse Laplace transforms of functions F(s). Our first step will be to identify a transform pair f(t) and F(s) for illustration in the development.

1. Let f be the function given by
   

   Use your computer algebra system to verify that the Laplace transform F of f is given by

   

We begin our search for an inversion procedure for the Laplace transform by looking at the inversion process that we know for the Fourier transform. Suppose g is an absolutely integrable function on the real line. Then we know that

Now our test function f grows like t² as t approaches infinity. So for any positive number a, the function g defined by

is absolutely integrable. (Here we use the fact that, for positive values of t, the negative exponential "kills" the polynomial growth -- and g is zero for negative values of t.) Thus, for positive t,

Now multiply both sides of this equation by e^at to obtain

Compare the inside integral in this representation of f to the definition of F as the Laplace transform of f:

If we extend the definition of F to allow complex values for the independent variable, the inside integral in our representation for f is just F(a + i omega), and our formula for f becomes

2. The integrals
   

   converge very slowly. To see why, set a = 1 and t = 1 and plot

   

   as a function of omega.

3. Use your computer algebra worksheet to evaluate the integrals for b = 10, 20, and 100. Compare the calculations to f(t).

4. Repeat Steps 2 and 3 for a = 1.15 and t = 0.9.