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 > α cos βx s s 2 + β 2 > 0 sin β s 2 + β 2 > 0 e αx cos s - α ( s - α ) 2 + β 2 > α e αx sin β ( s - α ) 2 + β 2 > α x n ,n =1 , 2 ,... n ! s n +1 > 0 x n e αx , 2 n ! ( s - α ) n +1 > α x cos s 2 - β 2 ( s 2 + β 2 ) 2 > 0 x sin 2 βs ( s 2 + β 2 ) 2 > 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 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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This note was uploaded on 02/09/2011 for the course MATH 3321 taught by Professor Morgan during the Fall '08 term at University of Houston.

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

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