38 Modeling and Analysis of Dynamic Systems TABLE 2.2 Laplace Transform Pairs f(t) F(s) 1 2 Unit impulse δ(t) 1, Unit step us(t) 3 t, Unit ramp ur(t) 1 1 s 1 s2 4 5 6 7 δ(t − a) u(t − a) tn – 1, n = 1,2,3, … ta – 1, a > 0 8 e−at 9 te−at 1 (s + a) 2 10 tne−at, n = 1, 2, 3, … n! (s + a)n +1 11 1 (e − at − e − bt ) , a ≠ b b−a 1 (s + a)(s + b) 12 1 (ae − at − be − bt ) , a ≠ b a−b s (s + a)(s + b) 13 1 1 (be − at − ae − bt ) 1+ ab a − b 1 s(s + a)(s + b) 14 1 (−1 + at + e − at ) a2 1 s 2 (s + a) 15 1 (1 − e − at − ate − at ) a2 1 s (s + a) 2 16 sinωt 17 cosωt 18 e−σt sinωt 19 e−σt cosωt s+σ (s + σ ) 2 + ω 2 20 1 − cosωt ω2 s (s + ω 2 ) 21 ωt − sinωt ω3 s (s 2 + ω 2 ) 22 t cosωt s2 − ω 2 (s 2 + ω 2 ) 2 23 1 t sinωt 2ω s (s 2 + ω 2 ) 2 No. e–as e–as/s (n − 1)!/sn Γ(a)/sa 1 s+a ω s2 + ω 2 s s2 + ω2 ω (s + σ ) 2 + ω 2 2 2 (continued) 39 Complex Analysis, Differential Equations, and Laplace Transformation TABLE 2.2 (Continued) Laplace Transform Pairs f(t) F(s) 24 1 (sin ωt − ωt cos ωt ) 2ω 3 1 (s 2 + ω 2 ) 2 25 1 (sin ωt + ωt cos ωt ) 2ω s2 (s 2 + ω 2 ) 2 26 1 1 1 sin ω 2t − sin ω1t , ω12 ≠ ω 22 ω1 ω − ω12 ω 2 1 (s 2 + ω12 )(s 2 + ω 22 ) 27 1 (cos ω1t − cos ω 2t ), ω12 ≠ ω 22 ω 22 − ω12 s (s 2 + ω12 )(s 2 + ω 22 ) 28 sinh at a s2 − a2 29 cosh at s s2 − a2 30 1 1 1 sinh at − sinh bt , a ≠ b b a 2 − b 2 a 1 (s 2 − a 2 )(s 2 − b 2 ) 31 1 [cosh at − cosh bt ], a ≠ b a2 − b2 s (s 2 − a 2 )(s 2 − b 2 ) 32 1 3 at 1 − at 1 at − π e + 2e 2 sin 2 6 3a 2 1 s 3 + a3 33 1 3 at 1 − at 1 at + π −e + 2e 2 sin 3a 2 6 s s 3 + a3 34 1 3 − at 1 − at 1 e + 2e 2 sin at + π 2 2 6 3a 1 s 3 − a3 35 1 3 − at 1 − at 1 at − π e + 2e 2 sin 3a 2 6 s s 3 − a3 36 1 [cosh at sin at − sinh at cos at ] 4a3 1 s 4 + 4a 4 37 1 sinh at sin at 2a 2 s s 4 + 4a 4 38 1 (sinh at − sin at ) 2a 3 1 s4 − a4 39 1 (cosh at − cos at ) 2a 2 s s4 − a4 No. 2 2