Laplace Transform Table Properties and Rules Function Transform Already know. � ∞ f (t)e−st dt f (t) F (s) = a f (t) + b g(t) a F (s) + b G (s) (Linearity) F (s − z) (s-shift) zt e f (t) 0− (Deﬁnition) Will learn in a future session. f � (t) sF (s) − f (0− ) f �� (t) s 2 F ( s ) − s f (0− ) − f � (0− ) f (n) ( t ) sn F (s) − sn−1 f (0− )− t f (t) − F � (s) tn f (t) (−1)n F (n) (s) u(t − a) f (t − a) e−as F (s) u(t − a) f (t) ( f ∗ g)(t) = � t+ 0− f (τ ) dτ e � t+ 0− − as L( f (t + a)) f (t − τ ) g(τ ) dτ F (s) s � ∞ s (t-translation) (t-translation) F (s) G (s) Interesting, but not included in this course. f (t) t · · · − f ( n −1) (0− ) F (σ) dσ (The function table is on the next page.) (integration rule) Laplace Transform Table OCW 18.03SC Function Table Function Transform Region of convergence 1/s Re(s) > 0 Already know. 1 e at 1/(s − a) Re(s) > a 2 Re(s) > 0 t 1/s tn n!/sn+1 2 Re(s) > 0 2 cos(ωt) s/(s + ω ) sin(ωt) ω/(s2 + ω 2 ) zt Re(s) > 0 Re(s) > 0 2 2 e cos(ωt) (s − z)/((s − z) + ω ) Re(s) > Re(z) ezt sin(ωt) ω/((s − z)2 + ω 2 ) Re(s) > Re(z) δ(t) 1 all s δ(t − a) e−as all s s/(s2 − k2 ) Re(s) > k k/(s2 − k2 ) Re(s) > k cosh(kt) = ekt + e−kt 2 kt e − e−kt sinh(kt) = 2 Will learn in this session. 1 (sin(ωt) − ωt cos(ωt)) 2ω 3 t sin(ωt) 2ω ( s2 1 (sin(ωt) + ωt cos(ωt)) 2ω s2 ( s2 + ω 2 )2 Re(s) > 0 u(t − a) e−as /s Re(s) > 0 tn e at n!/(s − a)n+1 Re(s) > a 1 + ω 2 )2 s ( s2 + ω 2 )2 Re(s) > 0 Re(s) > 0 Will learn in a future session. Interesting, but not included in this course. 1 1 √ √ s πt ta Γ ( a + 1) s a +1 2 Re(s) > 0 Re(s) > 0