"[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"