Laplace Transform

An integral transform that maps a time-domain function f(t) defined for t ≥ 0 to a complex-frequency function F(s) via F(s) = ∫_0^∞ e^{-s t} f(t) dt, with a region of convergence in the complex s-plane.

An integral transform that maps a time-domain function f(t) (usually for t ≥ 0) to F(s) = ∫_0^∞ e^{-st} f(t) dt, producing a complex-frequency representation parametrized by s and widely used to solve linear ODEs and analyze system behavior.