1. 4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

2. Examples

3. 4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

4. Examples

7. Summary

8. Inverse Laplace

10. Translation Translation on the s-axis Translation on the t-axis

11. Heaviside function (Unit step fun) 1)Step function defined on non-negative x-axis (positive x + zero ) 2) f(t)u(t-a) turns off a portion of the graph of f 3)Can be used to write piecewise function in a compact form 1 t

12. Can be used to write piecewise function in a compact form

13. 1 t 2 Write in compact form

14. Translation Translation on the s-axis Translation on the t-axis

15. 1 t 2

16. Inverse Form

18. Alternative Form Translation on the t-axis We are frequently confronted with the problem of finding the Laplace Transform of a product of a function g and a unit step function u(t-a)

19. Convolution Remarks

20. Convolution

22. Convolution (Inverse Form)

23. Convolution (special case) 7/pp218 8/pp218

24. 4 types of equations 1) Algebraic 2) Differential 3) Integral 4) Integrodifferential

25. Volterra Integral Equation

26. Laplace of Periodic function Remark: The Laplace Transform of a periodic function can be obtained by integration over one period proof

27. Find the Laplace Transform of the periodic function shown in the figure. Example 7 t 1