MIT18_03S10_c26

MIT18_03S10_c26 - 18.03 Lecture 26 April 9 2010 Laplace...

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18.03 Lecture 26 , April 9, 2010 Laplace Transform: definition and basic properties 1. Definition of LT; L[1] 2. Region of convergence 3. Powers 4. Linearity 5. s-shift rule 6. sines and cosines 7. t-domain and s-domain [1] Laplace Transform We continue to consider functions f(t) which are zero for t < 0 . (I may forget to multiply by u(t) now and then.) The Laplace transform takes a function f(t) (of "time") and uses it to manufacture another function F(s) (where s can be complex). It: [Slide] (1) makes explicit long term behavior of f(t) . (2) answers the question: if I know w(t), how can I compute p(s) ? (3) converts differential equations into algebraic equations. But we won't see these virtues right away. Definition: The Laplace transform of f(t) is the improper integral F(s) = integral_{0}^¥infty e^{-st} f(t) dt (formula subject to two refinements). We will often write f(t) ------> F(s) and L[f(t)] = F(s) (This notation isn't so good, because there's no room for "s" on the left.) For each value of s , F(s) is a weighted integral of f(t). Wnen s = 0 , for example, the Laplace integral is just the integral of f(t). When s > 0 , the values of f(t) for large t are given less weight. Each value of F(s) contains information about the whole of f(t). Example:
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MIT18_03S10_c26 - 18.03 Lecture 26 April 9 2010 Laplace...

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