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Unformatted text preview: ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------18.03 Lecture 26, April 14Laplace Transform: basic properties; functions of a complex variable;polesdiagrams; s-shift law.[1] The Laplace transform connects two worlds: | The t domain|||| t is real and positive|||| functions f(t) are signals, perhaps nasty, with discontinuities|| and delta functions|||| ODEs relating them|||| convolution|||| systems represented by their weight functions w(t)|||| ^L | | L^{-1}v | | The s domain|||| s is complex|||| beautiful functions F(s) , often rational = poly/poly|||| and algebraic equations relating them|| ------------------------------------------------------------------------|| ordinary multiplication of functions|||| systems represented by their transfer functions W(s)||| The use in ODEs will be to apply L to an ODE, solve the resulting verysimple algebraic equation in the s using the "inverse Laplace transform"world, and then return to realityL^{-1}. [2] The definition can be motivated but it is more efficient to simplygive it and come to the motivation later. Here it is....
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