Table 1: Table of Laplace Transforms Number f (t) F (s) 1 δ(t) 2 us(t) 3 t 4 tn −at5 e6 te−at 1 tn−1 −at7 e(n−1)! −at81 − e9 e−at − e−bt 10 be−bt − ae−at 11 sinat 12 cosat 13 e−at cosbt 14 e−at sinbt 15 1 − e−at(cosbt + a b sinbt) 1 1 s 1 2sn! sn+1 1 (s+a) 1 (s+a)2 1 (s+a)n a s(s+a) b−a (s+a)(s+b) (b−a)s (s+a)(s+b) a 2s2+as 2s2+as+a (s+a)2+b2 b (s+a)2+b2 a2+b2 s[(s+a)2+b2] 1Table 1: Properties of Laplace Transforms Number Time Function Laplace Transform Property 1 αf1(t)+ βf2(t) αF1(s)+ βF2(s) Superposition 2 f (t − T )us(t − T ) F (s)e −sT ; T ≥ 0 Time delay 3 f (at) 1 a F ( s a ); a> 0 Time scaling 4 e −atf (t) F (s + a) Shift in frequency 5 6 df (t) dt d2f (t) dt2 sF (s) − f (0−) s 2F (s) − sf (0−) − f (1)(0−) First-order differentiation Second-order differentiation 7 f n(t) s nF (s) − s n−1f (0) − s n−2f (1)(0) − ... − f (n−1)(0) n th-order differentiation 6 Zt 0− f (ζ)dζ 1 s F (s) Integration 7 f (0+) lim s→∞ sF (s) Post-initial value theorem 8 lim t→∞ f (t) lim s→0 sF (s) Final value theorem 9 tf (t) − dF (s) ds Multiplication by time
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