1. “The wise are instructed by reason; ordinary minds by experience; the stupid, by necessity; and brutes, by instinct.” -Cicero “A man’s judgement is no better than his information.” - from Bits & Pieces

2. Laplace transforms Introduction Review of Complex Variables and Complex functions Laplace transformation Inverse Laplace transformation Partial Fraction Expansion with MATLAB Solving DEs Assignments (start now)

3. Introduction

4. Review of Complex Variables and Complex functions

5. Limits are path dependent. Consider 2 paths.

6. Review of Complex Variables and Complex functions Limits are path dependent. Consider 2 paths. If equal Equating real and imaginary parts Cauchy-Riemann Conditions. Only if these two conditions are satisfied, the function G(s) is analytic.

7. Review of Complex Variables and Complex functions Points in the s plane at which the function G(s) is analytic are called ordinary points. Points in the s plane at which the function G(s) is not analytic are called singular points. Singular points at which the function G(s) or its derivatives approach infinity are called poles. Singular points at which the function G(s) equals zero are called zeros. If G(s) approaches infinity as s approaches –p and if the function G(s)(s+p) n, for n = 1, 2, 3, … has a finite, nonzero value at s = -p, then s=-p is called a pole of order n. If n = 1, the pole is called a simple pole.

8. Review of Complex Variables and Complex functions Euler’s Theorem Corollaries Proof: Consider the Taylor series expansions of the functions

9. Laplace Transformation

12. Inverse Laplace Transformation Partial fraction Expansion. “Cover up Rule” A: B:B:

13. Laplace transform of a derivative Primes and dots are often used as alternative notations for the derivative. Dots are almost always used to denote time derivatives. Primes might denote either time or space derivatives. In problems with both time and space derivatives, primes are space derivatives and dots are time derivatives. Note: Lower case f indicates function of time. Upper case F indicates function of s. (Multiplication by s) = (differentiation wrt time)

14. Inverse Laplace Transformation A and B same as in previous problem.

15. (Today’s date: 8/29/03) Assignment due next class period You will receive two pieces of paper –One has “Good” (not necessarily perfect) responses to the quiz questions. –The other has a different response to the quiz questions. These may or may not have errors on them. This page has your name on a label. Use the notes and the “Good” responses to make corrections on the quiz responses on the page with your name on it.

16. Inverse Laplace Transformation

18. Partial Fraction Expansion with MATLAB Read Section 2-6 Feel free to use MatLab to check your work. You will not have access to MATLAB on tests.

19. Solving DEs Laplace transforms are the primary tool used to solve DEs in control engineering. When initial conditions are zero: For non zero initial conditions

20. Solving DEs

21. No j’s in final answer.

22. Assignment Read Chapter 2. Un-graded homework. Be able to work, without referring to the book, A-2-2 thru A-2-7, A-2-11 thru A-2-14, A-2-17. Graded homework, due next class. B-2-1, B-2-11, B-2-13, B-2- 14, B-2-18 thru 23. Quiz. Solve a differential equation similar to one in the assignments.

23. A2-1Poles A2-2 thru A2-4, A2-10Laplace transformB2-1 thru B2-6 A2-5 thru A2-7Laplace transform theorems B2-9 A2-8 thru A2-9proofs A2-11 thru A2-14Inverse Laplace transform B2-11 thru B2-14 A2-15 thru A2-16Inverse Laplace transform MATLAB B2-15 thru B2-17 A2-17Solve DEsB2-18 thru B2-23