Definition and key properties
The one-sided Laplace transform is:
\[F(s) = \mathcal{L}\{f(t)\}(s) = \int_0^\infty f(t)\,e^{-st}\,dt, \quad s \in \mathbb{C}\]Key properties include linearity, the differentiation rule $\mathcal{L}{f’}(s) = sF(s) - f(0)$, and the convolution theorem $\mathcal{L}{f * g} = F(s)G(s)$. Common pairs: $\mathcal{L}{e^{at}} = 1/(s-a)$, $\mathcal{L}{\sin(\omega t)} = \omega/(s^2+\omega^2)$.
Transfer functions and stability
For an LTI system governed by an ODE, taking the Laplace transform yields the transfer function $H(s) = Y(s)/U(s)$ as a ratio of polynomials in $s$. A system is bounded-input bounded-output stable if and only if all poles of $H(s)$ lie strictly in the left half-plane $\operatorname{Re}(s) < 0$. The inverse Laplace transform is recovered via the Bromwich contour integral or partial fraction decomposition.
Applications
The Laplace transform appears in electrical circuit analysis (impedance $Z(s) = V(s)/I(s)$), mechanical systems, PID controller design, and solving initial value problems. The Fourier transform is the special case $s = i\omega$ evaluated on the imaginary axis, valid when the ROC contains it.