7.1 notes

# 7.1 notes - 2 − k 2 L cosh kt = s s 2 − k 2 1 3 Su ﬃ...

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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 →∞ b a 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 ) } = 0 e st f ( t ) dt is said to be the Laplace transform of f , provided that the integral converges. In general, L{ f ( t ) } = F ( s ) , L{ g ( t ) } = G ( s ) . Ex: Find the Laplace transform. L{ 1 } , L{ t } , L{ e 2 t } , L{ sin 3 t } L is a linear transform: i.e. L{ α f ( t ) + β g ( t ) } = α F ( s ) + β G ( s ) So L{ 5 + 4 sin 3 t 7 e 2 t } = Theorem: Transforms of some basic functions. 1. L{ 1 } = 1 s . 2. L{ t n } = n ! s n +1 , n = 1 , 2 , 3 , . . . . 3. L{ e at } = 1 s a . 4. L{ sin kt } = k s 2 + k 2 , L{ cos kt } = s s 2 + k 2 . 5. L{ sinh kt } = k s
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Unformatted text preview: 2 − k 2 , L{ cosh kt } = s s 2 − k 2 . 1 3. Su ﬃ cient 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 > , and T > 0 such that | f ( t ) | ≤ Me ct 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 lim s →∞ F ( s ) = 0 . 2...
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