# Ch 2 Laplace Transforms

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### Text of Ch 2 Laplace Transforms

• The wise are instructed by reason;ordinary minds by experience;the stupid, by necessity; andbrutes, by instinct. -CiceroA mans judgement is no better than his information. - from Bits & Pieces

• Laplace transformsIntroduction

Review of Complex Variables and Complex functions

Laplace transformation

Inverse Laplace transformation

Partial Fraction Expansion with MATLAB

Solving DEs

Assignments (start now)

• Introduction

• Review of Complex Variables and Complex functions

• Review of Complex Variables and Complex functionsLimits are path dependent. Consider 2 paths.

• Review of Complex Variables and Complex functionsLimits are path dependent. Consider 2 paths. If equalEquating real and imaginary partsCauchy-RiemannConditions.Only if these two conditions are satisfied, the function G(s) is analytic.

• Review of Complex Variables and Complex functionsPoints 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.

• Review of Complex Variables and Complex functionsEulers TheoremCorollariesProof: Consider the Taylor series expansions of the functions

• Laplace Transformation

• Laplace Transformation

• Laplace Transformation

• Inverse Laplace TransformationPartial fraction Expansion.Cover up RuleA:B:

• Laplace transform of a derivativePrimes 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)

• Inverse Laplace TransformationA and B same as in previous problem.

• (Todays date: 8/29/03) Assignment due next class period

You will receive two pieces of paperOne 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.

• Inverse Laplace Transformation

• Inverse Laplace Transformation

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

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

• Solving DEs

• Solving DEsNo js in final answer.

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.

• A2-1PolesA2-2 thru A2-4, A2-10Laplace transformB2-1 thru B2-6A2-5 thru A2-7Laplace transform theoremsB2-9A2-8 thru A2-9proofsA2-11 thru A2-14Inverse Laplace transformB2-11 thru B2-14A2-15 thru A2-16Inverse Laplace transform MATLABB2-15 thru B2-17A2-17Solve DEsB2-18 thru B2-23

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