Laplace transform

Laplace transform is an integral transform that maps a function of time into a function of complex frequency, converting differential equations into algebraic equations and convolution into multiplication. In network analysis and signal processing, it transforms the time-domain constitutive relations of capacitors and inductors into simple impedance expressions in the s-domain, allowing the full power of complex analysis to be applied to circuit behavior. The transform is not merely a mathematical convenience; it is a change of basis that reveals the system's poles and zeros — its resonant frequencies and damping modes — as geometric features in the complex plane. The Fourier transform is a special case of the Laplace transform evaluated on the imaginary axis, and the relationship between the two illuminates why stable systems have poles in the left half-plane.