Table 12.1 LAPLACE TRANSFORM PAIRS - Purdue University

Table LAPLACE TRANSFORM PAIRS Item Number f(t) LLLL[f(t)] = F(s) 1 K (t) K 2 Ku(t) or K K s 3 r(t) 1s2 4 tnu(t) n!sn+1 5 e atu(t) 1 (s+a) 6 te atu(t) 1 (s+a)2 7 tne atu(t) n!(s+a)n+1 8 sin( t)u(t) s2+ 2 9 cos( t)u(t) ss2+ 2 10 e atsin( t)u(t) (s+a)2+ 2 11 e atcos( t)u(t) (s+a)(s+a)2+ 2 12 tsin( t)u(t) 2 s(s2+ 2)2 13 tcos( t)u(t) s2 2(s2+ 2)2 14 sin( t + )u(t) ssin( )+ cos( )s2+ 2 15 cos( t + )u(t) scos( ) sin( )s2+ 2 16 e at[sin( t) tcos( t)]u(t) 2 3[(s+a)2+ 2]2 17 te atsin( t)u(t) 2 s+a[(s+a)2+ 2]2 18 e atC1cos( t)+C2 C1a sin( t) u(t) C1s+C2s+a( )2+ 2 Table LAPLACE TRANSFORM PROPERTIES Property TRANSFORM Pair Linearity L[a1f1(t) + a2f2(t)] = a1F1(s) + a2F2(s) Time Shift L[f(t T)u(t T)] = e sTF(s), T > 0 Multiplication by t L[tf(t)u(t)] = ddsF(s) Multiplication by tn L[tnf(t)]=( 1)ndnF(s)dsn Frequency Shift L[e atf(t)] = F(s + a) Time Differentiation Lddtf(t) = sF(s) f(0 ) Second-Order Differentiation Ld2f(t)dt2 =s2F(s) sf(0 ) f(1)(0 )

Table 12.2 LAPLACE TRANSFORM PROPERTIES Property Transform Pair Linearity L[a1f1(t) + a2f2(t)] = a1F1(s) + a2F2(s) Time Shift L[f(t – T)u(t – T)] = e ...

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Transcription of Table 12.1 LAPLACE TRANSFORM PAIRS - Purdue University

1 Table LAPLACE TRANSFORM PAIRS Item Number f(t) LLLL[f(t)] = F(s) 1 K (t) K 2 Ku(t) or K K s 3 r(t) 1s2 4 tnu(t) n!sn+1 5 e atu(t) 1 (s+a) 6 te atu(t) 1 (s+a)2 7 tne atu(t) n!(s+a)n+1 8 sin( t)u(t) s2+ 2 9 cos( t)u(t) ss2+ 2 10 e atsin( t)u(t) (s+a)2+ 2 11 e atcos( t)u(t) (s+a)(s+a)2+ 2 12 tsin( t)u(t) 2 s(s2+ 2)2 13 tcos( t)u(t) s2 2(s2+ 2)2 14 sin( t + )u(t) ssin( )+ cos( )s2+ 2 15 cos( t + )u(t) scos( ) sin( )s2+ 2 16 e at[sin( t) tcos( t)]u(t) 2 3[(s+a)2+ 2]2 17 te atsin( t)u(t) 2 s+a[(s+a)2+ 2]2 18 e atC1cos( t)+C2 C1a sin( t) u(t) C1s+C2s+a( )2+ 2 Table LAPLACE TRANSFORM PROPERTIES Property TRANSFORM Pair Linearity L[a1f1(t) + a2f2(t)] = a1F1(s) + a2F2(s) Time Shift L[f(t T)u(t T)] = e sTF(s), T > 0 Multiplication by t L[tf(t)u(t)] = ddsF(s) Multiplication by tn L[tnf(t)]=( 1)ndnF(s)dsn Frequency Shift L[e atf(t)] = F(s + a) Time Differentiation Lddtf(t) = sF(s) f(0 ) Second-Order Differentiation Ld2f(t)dt2 =s2F(s) sf(0 ) f(1)(0 )

2 Nth-Order Differentiation Ldnf(t)dtn =snF(s) sn 1f(0 ) sn 2f(1)(0 ) K f(n 1)(0 ) Time Integration (i) Lf(q)dq t =F(s)s+f(q)dq 0 s (ii) Lf(q)dq0 t =F(s)s Time/Frequency Scaling L[f(at)] = 1aFsa