“The wise are instructed by reason ; ordinary minds by experience ; the stupid , by necessity ; and brutes , by

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

3. Introduction

4. Review of Complex Variables and Complex functions

5. Review of Complex Variables and Complex functions 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 Proof: Consider the Taylor series expansions of the functions Corollaries

9. Laplace Transformation

10. Laplace Transformation

11. Laplace Transformation

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

13. Note: Lower case f indicates function of time. Upper case F indicates function of s. Laplace transform of a derivative (Multiplication by s) = (differentiation wrt time) 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.

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

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

20. Solving DEs

21. Solving DEs 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.