Laplace transform

In mathematics, the Laplace transform is an integral transform named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/). It transforms a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering. F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {displaystyle F(s)=int _{0}^{infty }f(t)e^{-st},dt}     (Eq.1) F ( s ) = ∫ − ∞ ∞ e − s t f ( t ) d t {displaystyle F(s)=int _{-infty }^{infty }e^{-st}f(t),dt}     (Eq.2) f ( t ) = L − 1 { F } ( t ) = 1 2 π i lim T → ∞ ∮ γ − i T γ + i T ⁡ e s t F ( s ) d s {displaystyle f(t)={mathcal {L}}^{-1}{F}(t)={frac {1}{2pi i}}lim _{T o infty }oint _{gamma -iT}^{gamma +iT}e^{st}F(s),ds}     (Eq.3) In mathematics, the Laplace transform is an integral transform named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/). It transforms a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering. The Laplace transform is similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. Laplace transforms are usually restricted to functions of t with t ≥ 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory. The Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (often 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. So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814) and the integral form of the Laplace transform evolved naturally as a result . The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. Laplace's use of generating functions was similar to what is now known as the z-transform and he gave little attention to the continuous variable case which was discussed by Abel. The theory was further developed in the 19th and early 20th centuries by Lerch, Heaviside, and Bromwich. The current widespread use of the transform (mainly in engineering) 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 Doetsch to whom the name Laplace Transform is apparently due. The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form as solutions of differential equations but did not pursue the matter very far. Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form which some modern historians have interpreted within modern Laplace transform theory.