Basic Laplace Transform Formula
\[L\{f(t)\} = F(s) = \int_0^\infty e^{-st}
f(t) dt\]
Properties of Laplace Transform
\[\begin{aligned}
\textbf{Linearity Property:} & \quad A f_1(t) + B f_2(t)
\leftrightarrow A F_1(s) + B F_2(s) \\
\textbf{Frequency Shifting Property:} & \quad e^{s_0t} f(t)
\leftrightarrow F(s - s_0) \\
\textbf{Integration Property:} & \quad \int_0^t f(\lambda) d\lambda
\leftrightarrow \frac{1}{s} F(s) \\
\textbf{Multiplication by Time:} & \quad t f(t)
\leftrightarrow -\frac{d}{ds} F(s) \\
\textbf{Complex Shift Property:} & \quad f(t) e^{-at}
\leftrightarrow F(s + a) \\
\textbf{Time Reversal Property:} & \quad f(-t)
\leftrightarrow F(-s) \\
\textbf{Time Scaling Property:} & \quad
f\left(\frac{t}{a}\right) \leftrightarrow a F(as)
\end{aligned}\]
Laplace Transform Table
| Sl No. | \(\boldsymbol{f(t)}\) | \(\boldsymbol{L(f(t)) = F(s)}\) |
|---|---|---|
| 1 | 1 | \(\frac{1}{s}\) |
| 2 | \(t^n\) at \(t = 1,2,3,\ldots\) | \(\frac{n!}{s^{n+1}}\) |
| 3 | \(\sqrt{t}\) | \(\frac{\sqrt{\pi}}{2s^{3/2}}\) |
| 4 | \(\sin(at)\) | \(\frac{a}{s^2 + a^2}\) |
| 5 | \(t \sin(at)\) | \(\frac{2as}{(s^2 + a^2)^2}\) |
| 6 | \(\sin(at+b)\) | \(\frac{(s \sin(b) + a \cos(b))}{s^2 + a^2}\) |
| 7 | \(\sinh(at)\) | \(\frac{a}{s^2 - a^2}\) |
| 8 | \(e(at) \sin(bt)\) | \(\frac{b}{(s-a)^2 + b^2}\) |
| 9 | \(e(ct) f(t)\) | \(F(s-c)\) |
| 10 | \(f'(t)\) | \(sF(s) - f(0)\) |
| 11 | \(e(at)\) | \(\frac{1}{s - a}\) |
| 12 | \(t^p\) at \(p > -1\) | \(\frac{\Gamma(p+1)}{s^{p+1}}\) |
| 13 | \(t^{n-1/2}\) at \(n = 1,2,\ldots\) | \(\frac{(1 \cdot 3 \cdot 5 \ldots (2n-1))\sqrt{\pi}}{2n s^{n+1/2}}\) |
| 14 | \(\cos(at)\) | \(\frac{s}{s^2 + a^2}\) |
| 15 | \(t \cos(at)\) | \(\frac{s^2 - a^2}{(s^2 + a^2)^2}\) |
| 16 | \(\cos(at+b)\) | \(\frac{(s \cos(b) - a \sin(b))}{s^2 + a^2}\) |
| 17 | \(\cosh(at)\) | \(\frac{a}{s^2 - a^2}\) |
| 18 | \(e(at) \cos(bt)\) | \(\frac{(s-a)}{(s-a)^2 + b^2}\) |
| 19 | \(t^n f(t)\) at \(n = 1,2,3,\ldots\) | \((-1)^n F^{(n)}(s)\) |
| 20 | \(f''(t)\) | \(s^2F(s) - sf(0) - f'(0)\) |