# Laplace Transform

The Laplace transform is a widely used integral transform with many applications in physics and engineering. Denoted, it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s). It is named after Pierre-Simon Laplace, who introduced the transform in his work on probability theory.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In such analyses, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

### Other articles related to "laplace transform, transform, laplace":

Z-transform - Relationship To Laplace Transform - Process of Sampling
... Its one sided Laplace transform is defined as Now the Laplace transform of the sampled signal (discrete time) is called Star transform and is given by It can be seen that the Laplace transform ...
Laplace Transform Applied To Differential Equations
... The Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain ... The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions ... First consider the following property of the Laplace transform One by induction can prove that Now we consider the following differential equation ...
Stochastic Ordering - Other Stochastic Orders - Laplace Transform Order
... Laplace transform order is a special case of convex order where is an exponential function ... Clearly, two random variables that are convex ordered are also Laplace transform ordered ...
Laplace Transform - Examples: How To Apply The Properties and Theorems - Example 6: Determining Structure of Astronomical Object From Spectrum
... The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter ... and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible model of the distribution of matter in it (density ... information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement ...
Sumudu Transform - Relationship To Other Transforms
... The Sumudu transform is a simple variant of the Laplace transform which is also used in its so-called p-multiplied form (sometimes known as the Laplace–Carson ... is employed in Western countries, and the Laplace–Carson form remains the standard in Eastern Europe ... The Sumudu transform is thus a minor variant of form (3) in which p is replaced by 1/u and in this guise has been pressed into service for special purposes in the form shown in Equation (1) ...

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