1. Laplace Transform Applications of the Laplace transform
solve differential equations (both ordinary and partial)
application to RLC circuit analysis
Laplace transform converts differential equations in the time domain to algebraic equations in the frequency domain, thus 3 important processes:
(1) transformation from the time to frequency domain
(2) manipulate the algebraic equations to form a solution
(3) inverse transformation from the frequency to time domain

2. Definition of Laplace Transform
Definition of the unilateral (one-sided) Laplace transform
where s=+j is the complex frequency, and f(t)=0 for t<0
The inverse Laplace transform requires a course in complex variables analysis (e.g., MAT 461)

3. Singularity Functions
Singularity functions are either not finite or don't have finite derivatives everywhere
The two singularity functions of interest here are
(1) unit step function, u(t)
(2) delta or unit impulse function, (t)

4. Unit Step Function, u(t)
The unit step function, u(t)
Mathematical definition
Graphical illustration 1 t
u(t)

5. Extensions of the Unit Step Function
A more general unit step function is u(t-a)
The gate function can be constructed from u(t)
a rectangular pulse that starts at t= and ends at t=  +T
like an on/off switch 1 t a 1 t 
+T
u(t-) - u(t- -T)

6. Delta or Unit Impulse Function, (t)
The delta or unit impulse function, (t)
Mathematical definition (non-pure version)
Graphical illustration 1 t
(t) t0

7. Transform Pairs
The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here.

8. Laplace Transform Properties

9. Block Diagram Reduction

10. Block Diagram Reduction

11. Block Diagram Reduction

12. Block Diagram Reduction

16. Poles and Stability

17. Poles and Stability

18. Poles and Stability

19. Underdamped System (2nd Order)