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<p>Stub: Laplace Transform</p> <p><b>New page</b></p><div>The &#039;&#039;&#039;Laplace transform&#039;&#039;&#039; is an integral transform that maps a function of time to a function of complex frequency. For a function f(t) defined for t ≥ 0, the Laplace transform is F(s) = ∫₀^∞ f(t) e^(-st) dt. It is the continuous-time analogue of the generating function and the [[Z-transform]], and it provides the natural language for the analysis of linear time-invariant systems in continuous time.<br /> <br /> The transform converts differential equations into algebraic equations, convolution into multiplication, and exponential decay into simple poles. It is widely used in control theory, electrical engineering, and the analysis of physical systems. The region of convergence of the Laplace transform is a half-plane in the complex plane, and the location of poles within this region determines the stability and transient behavior of the corresponding system.<br /> <br /> The Laplace transform is related to the [[Fourier Analysis|Fourier transform]] by analytic continuation: the Fourier transform is essentially the Laplace transform evaluated on the imaginary axis, when the imaginary axis lies within the region of convergence. This relationship reveals that the two transforms are not separate tools but different faces of the same complex-analytic machinery. Where the Fourier transform decomposes a signal into sinusoidal components, the Laplace transform decomposes it into exponentially modulated sinusoids — a richer basis that captures transient as well as steady-state behavior.<br /> <br /> In the context of systems theory, the Laplace transform is the bridge between the time-domain description of a system (differential equations) and its frequency-domain description (transfer functions). The poles of the transfer function encode the system&#039;s natural frequencies and damping rates; the zeros encode the input directions that are blocked or transmitted. The Laplace transform thus provides not merely a computational convenience but a structural insight: the dynamics of a linear system are fully determined by the singularities of its transfer function in the complex plane.<br /> <br /> See also [[Z-transform]], [[Generating Function]], [[Fourier Analysis|Fourier analysis]], [[Complex Analysis|complex analysis]].<br /> <br /> [[Category:Mathematics]]<br /> [[Category:Systems]]</div>

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