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This integral transform has a number of properties that make it useful for analysing linear dynamical system s. The most significant advantage is that integration and differentiation become multiplication and division. (This is similar to the way that logarithms change multiplication of numbers to addition.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve. The inverse is the Bromwich integral, which is a complex integral.

Also, the output of a linear dynamic system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplicationArithmetic In its simplest form, multiplication is a quick way of adding identical numbers. The result of multiplying numbers is called a product''. The numbers being multiplied are called coefficients or factors and individually as the multiplicand and m, which often makes matters easier. For more information, see control theoryThis article is about an engineering theory called control theory. There is also a sociological theory of deviant behavior that is called control theory. In engineering and mathematics, control theory deals with the behaviour of dynamical systems over tim.

The Laplace transform is named in honor of Pierre-Simon LaplacePierre-Simon Laplace ( March 23 1749 March 5 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplace's equation. He was a believer in causal determinism. The Laplacian differential operator, much relied-upon in.

A sometimes convenient abuse of notation, prevailing especially among engineers and physicists, writes this in the following form:

When one talks about the Laplace transform, one is generally referring to the unilateral version. There also exists a bilateral Laplace transform, which is defined as follows:

The Laplace transform F(s) typically exists for all real numbers s > a, where a is a constant which depends on the growth behavior of f(t).

The Laplace transform can also be used to solve differential equationsThe use of Laplace transform makes it much easier to solve linear differential equations with given initial conditions. First consider the following relations: : : : Suppose we want to solve the given differential equation: : This equation is equivalent t and is used extensively in electrical engineeringElectrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. Its practitioners are called electrical engineer s. Electrical engineering is a broad field that encompasses many subfields..

An interesting aspect of Laplace transforms is that mathematicians to this day do not know its domain. In other words, there is no specific set of rules that one can check a function against to know if its Laplace transform can be taken.