Laplace Transform
4.9 Tables of Laplace Transforms
Table 4.1: Table of Laplace Transform
| [asciimath]f(t)[/asciimath] | [asciimath]F(s)=[/asciimath] [asciimath]\mathcal{L}{f}[/asciimath] | Domain of [asciimath]F(s)[/asciimath] |
| [asciimath]C[/asciimath] | [asciimath]C/s[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]t[/asciimath] | [asciimath]1/s^2[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]t^n,[/asciimath] [asciimath]n=1,2, ...[/asciimath] | [asciimath](n!)/s^(n+1)[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]sqrt(t)[/asciimath] | [asciimath]sqrt(pi)/(2s^(3//2))[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]e^(at)[/asciimath] | [asciimath]1/(s-a)[/asciimath] | [asciimath]s>a[/asciimath] |
| [asciimath]t^n\e^(at),[/asciimath] [asciimath]n=1,2, ...[/asciimath] | [asciimath](n!)/(s-a)^(n+1)[/asciimath] | [asciimath]s>a[/asciimath] |
| [asciimath]sin(bt)[/asciimath] | [asciimath]b/(s^2+b^2)[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]cos(bt)[/asciimath] | [asciimath]s/(s^2+b^2)[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]e^(at)sin(bt)[/asciimath] | [asciimath]b/((s-a)^2+b^2)[/asciimath] | [asciimath]s>a[/asciimath] |
| [asciimath]e^(at)cos(bt)[/asciimath] | [asciimath](s-a)/((s-a)^2+b^2)[/asciimath] | [asciimath]s>a[/asciimath] |
| [asciimath]tsin(bt)[/asciimath] | [asciimath](2bs)/(s^2+b^2)^2[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]tcos(bt)[/asciimath] | [asciimath](s^2-b^2)/(s^2+b^2)^2[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]sinh(bt)[/asciimath] | [asciimath]b/(s^2-b^2)[/asciimath] | [asciimath]s>abs(b)[/asciimath] |
| [asciimath]cosh(bt)[/asciimath] | [asciimath]s/(s^2-b^2)[/asciimath] | [asciimath]s>abs(b)[/asciimath] |
| Step Function: [asciimath]u_a(t)=u(t-a)[/asciimath] | [asciimath]e^(-as)/s[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]u(t-a)f(t-a)[/asciimath] | [asciimath]e^(-as) F(s)[/asciimath] | [asciimath]a>0[/asciimath] |
| Direct Delta Function: [asciimath]delta(t-a)[/asciimath] | [asciimath]e^(-as)[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]e^(at)f(t)[/asciimath] | [asciimath]F(s-a)[/asciimath] | [asciimath]s>0[/asciimath] |
| [asciimath]t^kf(t)[/asciimath] | [asciimath](-1)^kF^((k))(s)[/asciimath] | |
| [asciimath]int_0^tf(x)dx[/asciimath] | [asciimath](F(s))/s[/asciimath] | |
| [asciimath]int_0^tf(t-x)g(x)dx[/asciimath] | [asciimath]F(s).G(s)[/asciimath] |
Table 4.2: Properties of Laplace Transform
| Property | Example |
| [asciimath]\mathcal{L}{f+g}[/asciimath] [asciimath]=\mathcal{L}{f} +\mathcal{L}{g}[/asciimath] | [asciimath]\mathcal{L}{t+cos(2t)}[/asciimath] [asciimath]=\mathcal{L}{t}[/asciimath] [asciimath]+\mathcal{L}{cos(2t)}[/asciimath] [asciimath]=1/s^2+s/(s^2+2^2)[/asciimath] |
| [asciimath]\mathcal{L}{cf}[/asciimath] [asciimath]=c \mathcal{L}{f}[/asciimath] for any constant [asciimath]c[/asciimath] | [asciimath]\mathcal{L}{4t}[/asciimath] [asciimath]=4\mathcal{L}{t}[/asciimath] [asciimath]=4(1/s^2)[/asciimath] |
| [asciimath]\mathcal{L}{e^(at)f}(s)[/asciimath] [asciimath]=\mathcal{L}{f} (s-a)[/asciimath] | [asciimath]\mathcal{L}{e^(3t)sin(5t) }[/asciimath] [asciimath]=5/((s-3)^2+5^2)[/asciimath] |
| [asciimath]\mathcal{L}{f'}[/asciimath] [asciimath]=s\mathcal{L}{f} -f(0)[/asciimath] | |
| [asciimath]\mathcal{L}{f''}[/asciimath] [asciimath]=s^2\mathcal{L}{f} -s f(0)-f'(0)[/asciimath] | |
| [asciimath]\mathcal{L}{t^n f(t)}[/asciimath] [asciimath]=(-1)^n (d^n)/(ds^n)(\mathcal{L}{f})[/asciimath] | [asciimath]\mathcal{L}{t^1 sin(7t)}[/asciimath] [asciimath]=(-1)^1 (d)/(ds)(\mathcal{L}{sin(7t)})[/asciimath] [asciimath]=-d/(ds)(7/(s^2+7^2))[/asciimath] [asciimath]=(14s)/(s^2+49)^2[/asciimath] |
