The Laplace Transformation Formula is a mathematical operation used to transform a time-domain function into a complex frequency-domain function . This transformation is particularly useful in solving differential equations and analyzing systems in engineering and physics.
[latex]
\[
\textbf{Laplace Transformation Formula}
\]
\[
\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt
\]
\[
\text{Where:}
\]
\[
f(t) \text{ is the original function of time.}
\]
\[
F(s) \text{ is the Laplace transform of } f(t), \text{ a function of the complex variable } s.
\]
\[
s = \sigma + i\omega, \text{ where } \sigma \text{ is the real part and } \omega \text{ is the imaginary part.}
\]
\[
\textbf{Key Properties of Laplace Transformation}
\]
\[
\textbf{1. Linearity:}
\]
\[
\mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\}
\]
\[
\textbf{2. Differentiation in Time Domain:}
\]
\[
\mathcal{L}\{f'(t)\} = sF(s) – f(0)
\]
\[
\textbf{3. Integration in Time Domain:}
\]
\[
\mathcal{L}\left\{\int_{0}^{t} f(\tau) \, d\tau\right\} = \frac{F(s)}{s}
\]
\[
\textbf{4. Shifting Property:}
\]
\[
\mathcal{L}\{e^{at}f(t)\} = F(s-a)
\]