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c26 - 18.03 Lecture 26 April 14 Laplace Transform basic...

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------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ 18.03 Lecture 26, April 14 Laplace Transform: basic properties; functions of a complex variable; poles diagrams; 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 | |
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------------------------------------------------------------------------ | | 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 very simple algebraic equation in the s using the "inverse Laplace transform" world, and then return to reality L^{-1}. [2] The definition can be motivated but it is more efficient to simply give it and come to the motivation later. Here it is. We continue to consider functions (possibly generalized) f(t) such that f(t) = 0 for t < 0 .
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