Example 6.1.2

f[t_]=Piecewise[{{(t^2),0≤t<1},{(((t-1))^(-1)),1<t≤2},{1,2<t≤3}}]

Plot[f[t],{t,0,3}]

Not continuous on the interval.

Example 6.1.5

LaplaceTransform[t,t,s]

LaplaceTransform[(t^2),t,s]

LaplaceTransform[(t^n),t,s]

Example 6.1.6

LaplaceTransform[Cos[a*t],t,s]

Example 6.1.17

LaplaceTransform[t*Cosh[a*t],t,s]

Example 6.2.5

InverseLaplaceTransform[(2s+2)/(s^2+2s+5),s,t]

ComplexExpand[%]

Simplify[%]

Example 6.2.10

InverseLaplaceTransform[(2s-3)/(s^2+2s+10),s,t]

ComplexExpand[%]

Simplify[%]

Example 6.2.27 (c)

Integrate[t^(2n) Exp[-s t],{t,0,Infinity}]

(∑_(n=0)^∞)(((((-1))^n)(2n)!)/((2^(2n))(((n!))^2)(s^(2n+1))))

Integrate[t^(n) Exp[-s t],{t,0,Infinity}]

(∑_(n=0)^∞)(((((-1))^n)(n)!)/((2^(2n))(((n!))^2)(s^(n+1))))

(∑_(n=0)^∞)((((-1))^n)/(n!(2^(2n))(s^(n+1))))

Let's check our results:

LaplaceTransform[BesselJ[0,t],t,s]

(∑_(n=0)^∞)((((-1))^n)/((2^(2n))(((n!))^2)))LaplaceTransform[(t^(2n)),t,s]

LaplaceTransform[BesselJ[0,(Sqrt[t])],t,s]

(∑_(n=0)^∞)((((-1))^n)/((2^(2n))(((n!))^2)))LaplaceTransform[(t^n),t,s]