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Laplace Transform: Examples

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)}.Fact A:We haveL{eatf(t)}=F(s a).Fact B (Magic):Derivatives int Multiplication bys(well, almost).L{f (t)}=(s1) (F(s) f(0))=sF(s) f(0)L{f (t)}= s2s1 F(s) f(0) f (0) =s2F(s) sf(0) f (0)L{f(n)(t)}= snsn F(s) f(0) f(n 2)(0) f(n 1)(0) =snF(s) sn 1f(0) sf(n 2)(0) f(n 1)(0).

that has positive radius of convergence R>0. Analytic functions are the best-behaved functions in all of calculus. For example, every analytic function is in nitely-di erentiable: Theorem: Let f(x) be analytic at x 0, say f(x) = X1 n=0 a n(x x 0)nwith radius of convergence R>0. Then: (a) fis in nitely-di erentiable on the interval (x 0 R;x 0 + R).

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