102_1_W08-102-Week-3-Recap - s = 0 sinh at | a s 2 − a 2...

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12 NHAN LEVAN WEEKS: 3 RECAP 10. Laplace Transforms: Definition (10.1) L s { f ( t ) } := Z 1 0 e st f ( t ) dt := F ( s ) where f ( t ) is 0 for t < 0 — hence we could have written f ( t ) U ( t ) — and s is a complex variable. L s { f ( t ) } if it exists is a function of s . The part of the s -plane over which F ( s ) exists is called the DOC – Domain Of Convergence — of F ( s ). FACT . L s is a Linear Transformation: L s { af 1 ( t ) + bf 2 ( t ) } = a L s { f 1 ( t ) } + b L s { f 2 ( t ) } In the next page we list Laplace Transforms of simple signals and some basic properties of L s {·} which will be needed later.
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13 11. Laplace Transforms of Simple Signals In the following table you will see time-functions on the left and their Laplace Transforms on the right. f ( t ) | L s { f ( t ) } U ( t ) | 1 s , Re s > 0 , single (‘order 1’) pole at s = 0 δ ( t ) | 1 e at | 1 s + a , Re s > Re a, single pole at s = a e at | 1 s a , Re s > Re a, single pole at s = a a : real or complex t | 1 s 2 , Re s > 0 , double (‘order 2’) pole at s = 0 t n , n > 0 , | n ! s n +1 , Re s > 0 , pole of order n + 1 at s = 0 sin ω 0 t | ω 0 s 2 + ω 2 0 , Re s > 0 , single poles at s = ± i ω 0 cos ω 0 t | s s 2 + ω 2 0 , Re s > 0 , single poles at s = ± i ω 0 and single zero at
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Unformatted text preview: s = 0 sinh at | a s 2 − a 2 , Re s > a, single poles at s = ± a cosh at | s s 2 − a 2 , Re s > a, single poles at s = ± a and single zero at s = 0 More Properties : e ∓ βt f ( t ) | F ( s ± β ) , F ( s ) := L s { f ( t ) } f ( t − t d ) , t d > | e − st d F ( s ) , F ( s ) := L s { f ( t ) } Z t f ( τ ) dτ | 1 s F ( s ) , F ( s ) := L s { f ( t ) } t f ( t ) | − d ds F ( s ) , F ( s ) := L s { f ( t ) } t n f ( t ) , n ≥ | (for you to fill in) d dt f ( t ) | sF ( s ) − f (0) , F ( s ) := L s { f ( t ) } d 2 dt 2 f ( t ) | s 2 F ( s ) − sf (0) − ˙ f (0) , d n dt n f ( t ) | s n F ( s ) − s n − 1 f (0) − s n − 2 ˙ f (0) · · · − f n − 1 (0) , n ≥ 1...
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