Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace. It takes a function of a real variable (often time) to a function of a complex variable (complex frequency). It 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. 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 many applications in the sciences and technology.
- [T]he so-called Mellin transform... finds applications in finding the sum of infinite series, the asymptotic value of an integral involving a large parameter, signal analyis, and imaging technique. ...In probability theory, [it] is an important tool in studying the distributions of products of two random variables. In particular, the Mellin transform of the product of two independent random variables equals the product of the Mellin transforms of the two variables. The Mellin transform is closely related to the two-sided Laplace transform.
The so-called Mellin transform has been considered by Laplace and used by Riemann in his study of the zeta function. It was, however, Mellin who provided a systematic formation of the transform and its application to solve ODEs and to estimate the value of integrals. ...Hjalmar Mellin ...was a student of Mittang-Leffler and Weierstrass. The kernel for the Mellin transform is
,where is a constant that lies on the right of all singularities of the kernel function. With the proper change of variables, the Mellin transform can be converted to a two-sided Laplace transform. In particular, a two-sided Laplace transform can be written as
- K. T. Chau, Theory of Differential Equations in Engineering and Mechanics (2017) p. 659.
- Doetsch and Bernstein, beginning from the early 1920s, worked together on the subject of Laplace transformation, integral equations and convolutions. They published several papers together, in which the connection between the Laplace transformation and convolution, i.e., Faltung, is discussed often. ...[T]he Laplace transformation of a function , denoted by , where is defined for all real numbers , is the following complex function of :
.The relation between the Laplace transformation and convolution is...: ....In 1922, they remark, regarding the Laplace transformation: "[w]e distinguish the functions of a subfield and a field [Oberkörper], which are connected by a certain process. The operations in the subfield are actual, proper [eigentliche] ones, which are only symbolic in the field, but which in certain cases are capable of an actual analytical representation."
- Michael Friedman, A History of Folding in Mathematics: Mathematizing the Margins (2018) Science Networks Historical Studies 59, p. 351.
- Because the links between a convolution integral and a Laplace or Fourier transform are so important... we briefly present Borel's (1899) work on a "Laplace like" transform. Note Mellin's work (1896)... was unknown to Borel... Borel defined two functions and by their following Laplace integrals...:
- Roger Godard, "The Convolution as a Mathematical Object" (2017) Research in History and Philosophy of Mathematics p. 207.