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Table of Laplace Transforms t" C f(t) e E 1 at 1,2,... sin bt eat, n=1,2,... at cos bt sin bt cos bt F(s) = £{f}(s) S 1 s-a S n+1 S b 2 s+b S> 0 F s-a 0 0 S>0 " (s-a)² + b² (s-a)² + b² Spa spa s>a
Properties of Laplace Transforms £{f+g} = £{f} + £{g} L{cf} = c£{f} for any constant c at L{e ª¹f(t)} (s) = £{f}(s – a) L {f} (s) = s£{f}(s) - f(0) L {f''} (s) = s² £{f}(s) – sf(0) - f'(0) n £ {f(")} (s) = s"\x{f}(s) — s"¯ ¹f(0) – s^¯²f′(0) - ... – f(n − ¹) (0) d' L{t"f(t)} (s) = ( − 1)" —¯¯¯¯ (L{f}(s)) - n ds 2-1 {F₁+F₂} = 1¯¹ {\ {F₁} + + £¯ ¹ (F₂} £¯ ¹ {cF} = c£¯¹ {F}