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Laplace Transform: Examples - Stanford University

Laplace transform : ExamplesDef:Given a functionf(t) defined fort >0. ItsLaplace transformis thefunction, denotedF(s) =L{f}(s), defined by:F(s) =L{f}(s) = 0e stf(t)dt.(Issue:The Laplace transform is an improper integral. So, does it always exist? : Is thefunctionF(s) always finite?Answer:This is a little subtle. We ll discuss this next time.)Fact (Linearity):The Laplace transform islinear:L{c1f1(t) +c2f2(t)}=c1L{f1(t)}+c2L{f2(t)}. example 1:L{1}=1sExample 2:L{eat}=1s aExample 3:L{sin(at)}=as2+a2 example 4:L{cos(at)}=ss2+a2 example 5:L{tn}=n!sn+1 Useful Fact:Euler s Formula says thateit= cost+isinte it= cost isintTherefore,cost=12(eit+e it),sint=12i(eit e it). Laplace transform : Key PropertiesRecall:Given a functionf(t) defined fort >0. ItsLaplace transformisthe function, denotedF(s) =L{f}(s), defined by:F(s) =L{f}(s) = 0e stf(t) :In the following, letF(s) =L{f(t)}.

Laplace Transform: Existence Recall: Given a function f(t) de ned for t>0. Its Laplace transform is the function de ned by: F(s) = Lffg(s) = Z 1 0 e stf(t)dt: Issue: The Laplace transform …

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