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In mathematics the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace (). It takes a function of a positive real variable ''t'' (often time) to a function of a complex variable ''s'' (frequency). The Laplace transform is very 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 . A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable ''s''. So unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Also techniques of complex variables can be used directly to 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'' (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. It has applications in classical control theory, mechanical and electrical engineering, and physics. == History == The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called z transform) in his work on probability theory. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich. 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. These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. However, in 1785, Laplace took the critical step forward when, rather than just 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. He used an integral of the form: : akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He 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 also 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Laplace transform」の詳細全文を読む スポンサード リンク
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