Laplace transform

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:

int_{0^-}^infty e^{-st} f(t),dt.

The lower limit of 0^- is short notation to mean lim_{epsilon ightarrow +0} -epsilon and assures the inclusion of the entire dirac delta function delta (t) at 0 if there is such an impulse in f(t) at 0.

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The parameter s is in general complex:

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:s = sigma + i omega. ,

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

Related Topics:
Integral transform - Dynamical system - Differentiation - Integration - Logarithm - Integral equation - Differential equation - Polynomial equation

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The Laplace transform is named in honor of Pierre-Simon Laplace, who used the transform in his work on probability theory. The Laplace transform was discovered originally by Leonhard Euler.

Related Topics:
Pierre-Simon Laplace - Probability theory - Leonhard Euler

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Region of convergence

The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on the growth behavior of f(t), whereas the two-sided transform is defined in a range

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a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence. In the two-sided case, it is sometimes called the strip of convergence.

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There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.

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Inverse Laplace transform

The inverse Laplace transform is the Bromwich integral, which is a complex integral given by:

Related Topics:
Bromwich integral - Complex

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: f(t) = mathcal{L}^{-1} left{F(s) ight}

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