L16 - The Laplace Transform Definition Let f t be a...

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Unformatted text preview: The Laplace Transform Definition: Let f ( t ) be a function defined on the half–line [0 , ∞ ) , the Signal < Function > assume that f is piecewise continuous and does not grow faster than all polynomials at ∞ . The Laplace Transform of f is de- fined as L{ f ( t ) } ( s ) def = R ∞ e- st f ( t ) dt def = lim A →∞ R A e- st f ( t ) dt , for s > . We often write F ( s ) for L{ f ( t ) } ( s ) and G ( s ) for L{ g ( t ) } ( s ) , etc. Consider F ( s ) an encoding of f ( t ) . Some Facts 1) The Laplace transform is linear: L{ c 1 f 1 + c 2 f 2 } = c 1 L{ f 1 } + c 2 L{ f 2 } . 2) The Laplace transform is one–to–one: L{ f } = L{ g } = ⇒ f = g . Examples: a) f ( t ) = t = 1 . L{ t } ( s ) = lim A →∞ R A e- st t dt = lim A →∞- 1 s e- st t = A t =0 =- 1 s lim e- sA | {z } →- e- s | {z } =1 ! = 1 s : L{ 1 } ( s ) = 1 s . 2 b) f ( t ) = t n for n > . R t n e- st dt =- 1 s t n e- st + n s R t n- 1 e- st dt = ⇒ R A t n e- st dt =- 1 s t n e- st A + n s R A t n- 1 e- st dt = ⇒ R ∞ t n e- st dt = n s R ∞ t n- 1 e- st dt = ⇒ L{ t n } ( s ) = n s · L{ t n- 1 } ( s ) . c) Applying this with n = 1 , 2 , 3 ,... gives L{ t } ( s ) = 1 s , L{ t 1 } ( s ) = 1 s 1 s = 1 s 2 , L{ t 2 } ( s ) = 2 s 1 s 2 = 2 s 3 , L{ t 3 } ( s ) = 3 s 2 s 3 = 3!...
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L16 - The Laplace Transform Definition Let f t be a...

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