laplace - Laplace Transform Pairs F (s) = 0 f (t)est dt ,...

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Laplace Transform Pairs F ( s )= ± 0 - f ( t ) e - st dt , s C Number f ( t ) F ( s ) Poles 1 δ ( t ) 1 none 2 1 1 s simple 3 e - at 1 s + a simple 4 te - at 1 ( s + a ) 2 multiple 5 t n e - at n ! ( s + a ) n +1 multiple 6 cos( ωt ) s s 2 + ω 2 imaginary 7 sin( ) ω s 2 + ω 2 imaginary 8 e - at cos( ) s + a ( s + a ) 2 + ω 2 complex 9 e - at sin( ) ω ( s + a ) 2 + ω 2 complex
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Laplace Transform Properties Property Name L { af ( t )+ bg ( t ) } = aF ( s bG ( s ) linearity L { e - at f ( t ) } = F ( s + a ) frequency shift L { f ( t - T ) u s ( t - T ) } = e - sT F ( s ) time shift L { f ( at ) } = 1 a F ± s a ² scaling L { ˙ f ( t ) } = sF ( s ) - f (0 - ) Frst derivative L { ¨ f ( t ) } = s 2 F ( s ) - sf (0 - ) - ˙ f (0 - ) second derivative L { f (
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laplace - Laplace Transform Pairs F (s) = 0 f (t)est dt ,...

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