28 Facts About Laplace transform

In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable.

The transform has many applications in science and engineering because it is a tool for solving differential equations.

Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory.

Current widespread use of the Laplace transform came about during and soon after World War II, replacing the earlier Heaviside operational calculus.

The advantages of the Laplace transform had been emphasized by Gustav Doetsch, to whom the name Laplace transform is apparently due.

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However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular.

Laplace transform then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.

Laplace transform recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space, because those solutions were periodic.

In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.

Laplace transform of a function, defined for all real numbers, is the function, which is a unilateral transform defined by.

The Laplace transform is defined and injective for suitable spaces of tempered distributions.

The inverse Laplace transform is given by the following complex integral, which is known by various names :.

In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table, and construct the inverse by inspection.

The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory.

The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former but not in the latter sense.

The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem.

Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems.

The Laplace transform turns integral equations and differential equations to polynomial equations, which are much easier to solve.

Laplace transform can be viewed as a continuous analogue of a power series.

In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter is replaced by the continuous parameter, and is replaced by.

The Laplace transform is usually restricted to transformation of functions of with.

The generalized Borel Laplace transform allows a different weighting function to be used, rather than the exponential function, to Laplace transform functions not of exponential type.

Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject.

Unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function,.

Laplace transform is often used in circuit analysis, and simple conversions to the -domain of circuit elements can be made.

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Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal.

Laplace transform can be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering.

The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.