Laplacetransformed

Laplacetransformed refers to the result of applying the Laplace transform to a time-domain function. The Laplace-transformed function, usually denoted F(s), encodes the same information as the original function f(t) but in a complex frequency domain. The transform is defined by F(s) = ∫0^∞ e^{-st} f(t) dt, where s is a complex number s = σ + iω and the integral converges in a region of the complex plane called the region of convergence. The function f(t) is typically assumed to be piecewise continuous on [0, ∞) and of exponential order to guarantee convergence.

The Laplace transform is a linear operator: L{af + bg} = aL{f} + bL{g}. It is convenient for transforming differential equations into algebraic equations. Common Laplace-transformed pairs include L{1} = 1/s, Re(s) > 0; L{t^n} = n!/s^{n+1}; L{e^{at}} = 1/(s − a); L{cos(bt)} = s/(s^2 + b^2); L{sin(bt)} = b/(s^2 + b^2). Differentiation in time converts to multiplication in the s-domain: L{f'(t)} = sF(s) − f(0); higher derivatives follow similarly. Other properties include time-shifting, L{f(t − a)u(t − a)} = e^{−as}F(s}, scaling and frequency shifting: L{f(at)} = (1/|a|)F(s/a) for a > 0, and L{e^{at}f(t)} = F(s − a).

The inverse transform recovers f from F via f(t) = (1/2πi) ∫γ−i∞^{γ+i∞} e^{st} F(s) ds, known as the Bromwich integral, or more practically through transform tables and analytic inversion methods. The Laplace transform is widely used to solve linear ordinary differential equations, analyze linear time-invariant systems, circuit analysis, and control theory, where it provides a powerful bridge between time-domain behavior and frequency-domain representation.