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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
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Laplace transformsIntroduction
Review of Complex Variables and Complex functions
Laplace transformation
Inverse Laplace transformation
Partial Fraction Expansion with MATLAB
Solving DEs
Assignments (start now)
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Introduction
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Review of Complex Variables and Complex functions
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Review of Complex Variables and Complex functionsLimits are path
dependent. Consider 2 paths.
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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.
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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.
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Review of Complex Variables and Complex functionsEulers
TheoremCorollariesProof: Consider the Taylor series expansions of
the functions
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Laplace Transformation
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Laplace Transformation
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Laplace Transformation
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Inverse Laplace TransformationPartial fraction Expansion.Cover
up RuleA:B:
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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)
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Inverse Laplace TransformationA and B same as in previous
problem.
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(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.
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Inverse Laplace Transformation
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Inverse Laplace Transformation
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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.
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Solving DEsLaplace transforms are the primary tool used to solve
DEs in control engineering.When initial conditions are zero:For non
zero initial conditions
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Solving DEs
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Solving DEsNo js in final answer.
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AssignmentRead 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.
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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
