math2 - 1 Chapter 0 – Laplace Transform Laplace Transform...

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Unformatted text preview: 1 Chapter 0 – Laplace Transform Laplace Transform For a given function f ( t ) ( t ≥ 0), we multiply it by exp (- st ) and then integrate, that is F ( s ) = ∫ ∞ − dt ) t ( f ) st exp( (0-1) We can call such an operation as the Laplace Transform and F ( s ) = L ( f ) = ∫ ∞ − dt ) t ( f ) st exp( (0-2) Similarly, we can define the inverse of the Transform, that is Inverse Laplace Transform, by f ( s ) = L-1 ( F ) (0-3) Linearity : L (a f + b g ) = a L ( f ) + b L ( g ) (0-4) Exercise 1 Prove Equation (0-4) Solution - First Shifting Theorem : L ( exp (a t ) f ) = F( s- a) (0-5) Exercise 2 Prove Equation (0-5) 2 Solution - Transforms of derivatives Now let consider the transform of the first derivative of f , L ( f’ ) = ∫ ∞ − dt ) t ( ' f ) st exp( = exp (- st ) f ( t ) ∫ ∞ ∞ − + o o dt ) t ( f ) st exp( s Therefore, L ( f’ ) = s L ( f ) - f (0) (0-6) One can show readily that L ( f’’ ) = s 2 L ( f ) – s f (0) – f’ (0) (0-7) Exercise 3 Show that...
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math2 - 1 Chapter 0 – Laplace Transform Laplace Transform...

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