Laplace transform

Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace. It takes a function of a real variable $t$ (often time) to a function of a complex variable $s$ (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.

Quotes

[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 analysis, 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 $K(s,t)=t^{s-1}$ The Mellin transform and its inversion are defined as: $F(s)=M[f(x)]=\int \limits _{0}^{\infty }x^{s-1}f(x)\,dx$ ,
$f(x)=M^{-1}[F(s)]={\frac {1}{2\pi i}}\int \limits _{c-i\infty }^{c+i\infty }x^{-s}F(s)\,ds$ where $c$ 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 $L[g(t)]=\int \limits _{-\infty }^{+\infty }g(t)e^{-st}\,dt$
- 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 $f(t)$ , denoted by ${\mathcal {L}}(f)$ , where $f$ is defined for all real numbers $t>0$ , is the following complex function of $F$ : $F(t)={\mathcal {L}}(f)=\int \limits _{0}^{\infty }e^{-tu}f(u)\,du$ .The relation between the Laplace transformation and convolution is...: ${\mathcal {L}}(f*g)={\mathcal {L}}(f)\cdot {\mathcal {L}}(g)$ . ...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 $f(z)$ and $g(z)$ by their following Laplace integrals...: $f(z)=\int \limits _{0}^{+\infty }F(u)e^{-u/z}\,{\frac {du}{z}};\quad g(z)=\int \limits _{0}^{+\infty }G(v)e^{-v/z}\,{\frac {dv}{z}}$ and then showed that the convolution integral is $H(x)=\int \limits _{0}^{x}F(t)G(x-t)\,dt$ . The Laplace transform of the convolution integral $H(x)$ reduced to a simple product of the two separate transforms $f(z)$ and $g(z)$ . Borel failed to see all the possibilities of his theorem. Volterra... also did not see the possible uses... But... in 1920, Doetsch produced a doctoral thesis... on Borel's summability theory of diverging series. Doetsch knew Borel's proof and was able to introduce modern, proper mathematical ideas on convolution integrals and Laplace transforms. The word Faltung was first introduced by Doetsch and Bernstein in 1920. The Laplace transform... and the Fourier transform... are both adequate tools for evaluating a convolution integral. ...Doetsch would introduce the convolution integral by analogy with a Cauchy product between two power series...
- Roger Godard, "The Convolution as a Mathematical Object" (2017) Research in History and Philosophy of Mathematics p. 207.