Laplace transform

"[T]he so-called ... 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 , [it] is an important tool in studying the distributions of products of two s. 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. ... ...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)\, dswhere 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 asL[g(t)] = \int\limits_{-\infty}^{+\infty} g(t)e^{-st}\, dt"

"Doetsch and Bernstein, beginning from the early 1920s, worked together on the subject of Laplace transformation, integral equations and s. 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.""

"Because the links between a convolution integral and a Laplace or 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 between two ..."