# Lecture 21

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Laplace Transforms Improper Integrals, Piecewise Functions, and Integral Transforms

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MATLAB Quiz Thursday, March 13 Same Time As Your Section (APM B432) Alternate Time Sign Ups Are Online DON’T FORGET!!!! (NO MAKEUPS)
Laplace Transforms “Change of Variables” For Ordinary Differential Equations Need to review 2 ideas from Calculus

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Review Topic 1: Improper Integrals An Integral with Infinity In Limit is Called “Improper” Z 1 0 f ( t ) dt Evaluated with a limit lim C ! 1 Z C 0 = f ( t ) dt Two Behaviors: Converge Diverge Integral Goes To A Finite Constant Integral Goes To 1
Review Topic 2: Piecewise Continuous Functions A Function is Called “Piecewise Continuous” On An Interval [a,b] if: The interval can be divided into sub intervals , , t n < t < b , t 0 < t < t 1 a < t < t 0 t i < t < t i +1 And the function is continuous on each interval And lim t ! t i + f ( t ) lim t ! t i - f ( t ) Both exist (BUT DON’T HAVE TO EQUAL) for any t i

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Integral Transform Introducing a “Kernel Function” K ( s , t ) f ( t ) K ( s , t ) Z β dt = F ( s ) Now a function of s Is Given By An “Integral Transform” of a Function f ( t )
Integral Transform

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