Lecture 8: Inverse Laplace Transform

1
Lecture 8 Inverse Laplace Transform

• Case of a rational, strictly proper transform
• Simple poles
• Multiple poles
• Complex poles.
• Non-strictly proper case.

2
Inverse Laplace Transform
3
I. First, the strictly proper case
4
(No Transcript)
5
(No Transcript)
6
General case, simple roots.
Brute force method gives n equations, n unknowns.
7
Case 2 multiple roots.
8
(No Transcript)
9
Case 3 Complex roots.
The preceding methods apply also to complex
roots, simple or multiple, using complex
operations. Example
Already known from the table.
10
Complex roots, another method.
Since F(s) has typically real coefficients, the
complex roots come in conjugate pairs. Treating
these two roots at once, we can invert the
transform without complex operations.
Taking a second order polynomial to the form on
the right is called completing the square.
Implicitly it gives the roots. Example
11
(No Transcript)
12
(No Transcript)
13
Recapitulating
14
II. Proper (but not strictly proper) case
15
Improper case
strictly proper