Inverse Laplace Transform

A precise method for recovering time-domain functions using tables, algebra, shifting, and convolution.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

The inverse Laplace transform $\mathcal L^{-1}\{F(s)\}=f(t)$ recovers a time-domain function whose Laplace transform is $F(s)$. It is unique up to equality almost everywhere within standard classes such as piecewise-continuous functions of exponential order. Inversion usually uses known pairs and transform properties rather than evaluating the Bromwich integral.

Notation and mathematical language

Core pairs include $1/s\leftrightarrow1$, $1/(s-a)\leftrightarrow e^{at}$, $s/(s^2+\omega^2)\leftrightarrow\cos\omega t$, and $\omega/(s^2+\omega^2)\leftrightarrow\sin\omega t$. The shift $F(s-a)$ corresponds to $e^{at}f(t)$; a factor $e^{-as}$ corresponds to $u(t-a)f(t-a)$.

Conceptual picture

Partial fractions separate a rational transform into recognizable dynamical modes. Completing a square reveals damped sine and cosine terms. A delay factor does not merely shift the graph horizontally; the unit step also keeps the response zero before the delay.

Fully worked example

Interpretation and application

Inverse transforms solve linear initial value problems with impulses, switching, and delays. The recovered function exactly corresponds to the algebraic transform under its assumptions; an engineering input represented by an ideal step or impulse is itself an approximation to a physical signal.