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Unformatted text preview: Chapter 7. The laplace Transform §7.1 Deﬁnition of the Laplace Transform 1. Review of Integral with Inﬁnite Intervals: Let f be a function deﬁned on [a, ∞), then
∞ a f (t)dt = lim b→∞ a b f (t)dt provided that the limit exists and ﬁnite. 2. Laplace Transform: Let f be a function deﬁned for t ≥ 0. Then the integral ∞ L{f (t)} = e−st f (t)dt
0 is said to be the Laplace transform of f , provided that the integral converges. In general, L{f (t)} = F (s), Ex: Find the Laplace transform. L{1}, L{t}, L{e−2t }, L{sin 3t} L{g (t)} = G(s). L is a linear transform: i.e. L{αf (t) + β g (t)} = αF (s) + β G(s) So L{5 + 4 sin 3t − 7e−2t } = Theorem: Transforms of some basic functions. 1. L{1} = 1 . s 2. L{tn } = 3. L{eat } =
n! sn+1 , n 1 s−a . = 1, 2, 3, . . . . 4. L{sin kt} = k s2 +k 2 , k s2 −k 2 , L{cos kt} = s s2 +k 2 . s s2 −k 2 . 5. L{sinh kt} = L{cosh kt} =
1 3. Suﬃcient Conditions for Existence of L{f (t)}. Def: A function f is said to be of exponential order c if there exist constants c, M > 0, and T > 0 such that f (t) ≤ M ect for all t ≥ T. Theorem: If f is piecewise continuous on [0, ∞) and of exponential order c, then L{f (t)} exists for s > c. Theorem: If f is piecewise continuous on [0, ∞) and of exponential order and F (s) = L{f (t)}, then lims→∞ F (s) = 0. 2 ...
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This note was uploaded on 01/05/2011 for the course MATH 3310 taught by Professor Dr.du during the Fall '08 term at Kennesaw.
 Fall '08
 DR.DU
 Differential Equations, Equations

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