Table 1: Properties of Laplace Transforms

Table 1: Table of Laplace TransformsNumberf(t)F(s)1 (t)12us(t)1s3t1s24tnn!sn+15e at1(s+a)6te at1(s+a)271(n 1)!tn 1e at1(s+a)n81 e atas(s+a)9e at e btb a(s+a)(s+b)10be bt ae at(b a)s(s+a)(s+b)11 sinatas2+a212 cosatss2+a213e atcosbts+a(s+a)2+b214e atsinbtb(s+a)2+b215 1 e at(cosbt+absinbt)a2+b2s[(s+a)2+b2]1 Table 1: Properties of Laplace TransformsNumber Time FunctionLaplace TransformProperty1 f1(t)+ f2(t) F1(s)+ F2(s)Superposition2f(t T)us(t T)F(s)e sT;T 0 Time delay3f(at)1aF(sa);a>0 Time scaling4e atf(t)F(s+a)Shift in frequency5df(t)dtsF(s) f(0 )First-order differentiation6d2f(t)dt2s2F(s) sf(0 ) f(1)(0 )Second-order differentiation7fn(t)snF(s) sn 1f(0) sn 2f(1)(0) .. f(n 1)(0) nth-order differentiation6Zt0 f( )d 1sF(s)Integration7f(0+)lims sF(s)Post-initial value theorem8limt f(t)lims 0sF(s)Final value theorem9tf(t) dF(s)dsMultiplication by time2

Table 1: Table of Laplace Transforms Number f(t) F(s) 1 δ(t)1 2 us(t) 1 s 3 t 1 s2 4 tn n! sn+1 5 e−at 1 (s+a)6 te−at 1 (s+a)27 1 (n−1)!tn−1e−at 1 (s+a)n81−e−at a s(s+a) 9 e−at −e−bt b−a (s+a)(s+b)10 be−bt −ae−at (b−a)s (s+a)(s+b)11 sinat a s2+a2 12 cosat s s2+a2 13 e−at cosbt s+a (s+a)2+b214 e−at sinbt b (s+a)2+b215 1−e−at(cosbt+ a b sinbt) a2+b2 s[(s+a ...

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Transcription of Table 1: Properties of Laplace Transforms

1 Table 1: Table of Laplace TransformsNumberf(t)F(s)1 (t)12us(t)1s3t1s24tnn!sn+15e at1(s+a)6te at1(s+a)271(n 1)!tn 1e at1(s+a)n81 e atas(s+a)9e at e btb a(s+a)(s+b)10be bt ae at(b a)s(s+a)(s+b)11 sinatas2+a212 cosatss2+a213e atcosbts+a(s+a)2+b214e atsinbtb(s+a)2+b215 1 e at(cosbt+absinbt)a2+b2s[(s+a)2+b2]1 Table 1: Properties of Laplace TransformsNumber Time FunctionLaplace TransformProperty1 f1(t)+ f2(t) F1(s)+ F2(s)Superposition2f(t T)us(t T)F(s)e sT;T 0 Time delay3f(at)1aF(sa);a>0 Time scaling4e atf(t)F(s+a)Shift in frequency5df(t)dtsF(s) f(0 )First-order differentiation6d2f(t)dt2s2F(s) sf(0 ) f(1)(0 )Second-order differentiation7fn(t)snF(s) sn 1f(0) sn 2f(1)(0) .. f(n 1)(0) nth-order differentiation6Zt0 f( )d 1sF(s)Integration7f(0+)lims sF(s)Post-initial value theorem8limt f(t)lims 0sF(s)Final value theorem9tf(t) dF(s)dsMultiplication by time2

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Table 1: Properties of Laplace Transforms

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