Solving Inverse Laplace Transform by Pattern Matching
Solving the Bromwich integral requires integration in complex plane, which can be complicated. Though we can often evaluating inverse Laplace transform by pattern-matching with the table of inverse Laplace transform.
Example: What is the inverse Laplace transform of
Let , we have
And we can have
Applying the frequency shifting property,
Example: What is the inverse Laplace transform of ?
We know that
Let ⇒, and
Which means
Thus
Inverse Laplace Transforms and ROCs
Given an function in the s-plane, the inverse Laplace transform of the unilateral Laplace transform is unique. This uniqueness is due to the implicit assumption of causality in the unilateral transform, which restricts the region of convergence to a right-half plane in the s-plane.
By contrast, for a bilateral Laplace transform, the inverse Laplace transform is not unique and depend on the region of convergence.
Example: what is the inverse (bilateral) Laplace transform of ?
Expand with partial fractions:
For the pole at , we either have a function if or otherwise.
For the pole at , we either have a function if or otherwise.