Chapter 4 Slides - The Laplace Transform Let f be...

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The Laplace Transform Let f be continuous function on [0 , ). The Laplace transform of f , denoted by L [ f ( x )], or by F ( s ), is given by L [ f ( x )] = F ( s ) = 0 e - sx f ( x ) dx . The domain of F is the set of real numbers s for which the improper integral converges. 1
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f is of exponential order λ if there exists a positive number M and a nonnegative number A such that | f ( x ) | ≤ Me λx on [ A, ) . 2
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Examples: (a) Bounded functions, e.g., sin x, cos x (b) Powers of x : f ( x ) = x k . (c) Exponential fcns: f ( x ) = e ax . f ( x ) = e x 2 is not of exponential order. 3
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THEOREM: Let f be a continuous function on [0 , ). If f is of exponential order λ , then the Laplace trans- form L [ f ( x )] = F ( s ) exists for s > λ . 4
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f ( x ) F ( s ) = L [ f ( x )] 1 1 s , s > 0 e αx 1 s - α , s > α cos βx s s 2 + β 2 , s > 0 sin βx β s 2 + β 2 , s > 0 e αx cos βx s - α ( s - α ) 2 + β 2 , s > α e αx sin βx β ( s - α ) 2 + β 2 , s > α x n , n = 1 , 2 , . . . n ! s n +1 , s > 0 x n e αx , n = 1 , 2 , . . . n ! ( s - α ) n +1 , s > α x cos βx s 2 - β 2 ( s 2 + β 2 ) 2 , s > 0 x sin βx 2 βs ( s 2 + β 2 ) 2 , s > 0 5
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Properties of the Laplace Transform 1. L is a linear operator: L [ f 1 ( x ) + f 2 ( x )] = L [ f 1 ( x )] + L [ f 2 ( x )] L [ cf ( x )] = c L [ f ( x )] . 2. If f is continuously differen- tiable and of exponential order λ , then L [ f ( x )] exists for s > λ and L [ f ( x )] = s L [ f ( x )] - f (0) . 6
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If f is twice continuously differen- tiable with f and f of exponential order λ , then L [ f ( x )] exists for s > λ and L [ f ( x )] = s 2 L [ f ( x )] - sf (0) - f (0) . In general, if f, f , · · · , f ( n - 1) are of exponential order λ , then L [ f ( n ) ( x )] exists for s > λ and L [ f ( n ) ( x )] = s n L [ f ( x )] - s n - 1 f (0) - s n - 2 f (0) - · · · - f ( n - 1) (0) .
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