Table of Basic Laplace Transforms

Example: ramp function t

DOC and poles. DOC of the (one-sided) Laplace transform: to the right of the right-most pole (the one with greatest real part).

Properties of the Laplace Transform Example:

Another example:

Properties of the Laplace Transform (cont.) Postpone proofs, do examples.

Examples. 2)

Examples.

6) Delay property. No confusion if u(t) term is included in the original transform.

Initial and Final Value Theorems Relates limiting values of a function and its transform. Can be used to get time-domain information from the Laplace transform. Opposite limits in t and in s. Here we take s to be real-valued. SOME RESTRICTIONS APPLY! Precise statements given below, and proofs later. REMARKS:

Initial Value Theorem: Precise Statement Examples:

Final Value Theorem: Precise Statement Examples:

Inverse Laplace Transform

First, the strictly proper case

General case, simple roots. Brute force method gives n equations, n unknowns.