L74 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.4...

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Unformatted text preview: Fall 2003 Math 308/501502 7 Laplace Transforms 7.4 Inverse Laplace Transform Mon, 13/Oct c 2003, Art Belmonte Summary Theorem If f and g are continuous functions whose Laplace transforms are such that L { f } ( s ) = L { g } ( s ) for s > a, then f ( t ) = g ( t ) for all t > . Definition For a continuous function f of exponential order whose Laplace transform is F , we call f the inverse Laplace transform of F and write f = L- 1 { F } . (The preceding theorem is what makes this definition possible.) Linearity Property For constants and and inverse Laplace transforms L- 1 { F } = f and L- 1 { G } = g , we have L- 1 { F + G } = L- 1 { F } + L- 1 { G } = f + g Short table of [inverse] Laplace transforms # f ( t ) = L- 1 { F } ( t ) L { f } ( s ) = F ( s ) 1. 1 1 s , s > 2. t n n ! s n + 1 , s > 3. sin at a s 2 + a 2 , s > 4. cos at s s 2 + a 2 , s > 5. e at 1 s- a , s > a 6. e at sin bt b ( s- a ) 2 + b 2 , s > a 7. e at cos bt s- a ( s- a ) 2 + b 2 , s > a 8. t n e at n ! ( s- a ) n + 1 , s > a Notes When computing inverse Laplace transforms strictly with a pencil, we typically use partial fraction decomposition, completing the square, and algebraic manipulation before finally employing table lookup. On the other hand, the MATLAB Symbolic Math Toolbox (SMT) command ilaplace computes inverse Laplace transforms at...
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L74 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.4...

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