Meaning of Mellin transform | Babel Free

Definitions

An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.

Examples

“This chapter is concerned with the theory and applications of the Mellin transform⟳. We derive the Mellin transform⟳ and its inverse from the complex Fourier transform⟳. This is followed by several examples and the basic operational properties of Mellin transforms.”

“2005, Robb J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, page 303, If X is a positive random variable with density function f(x), the Mellin transform⟳ M(s) gives the (s-l)th moment of X. Hence Theorem 8.2.6 gives the Mellin transform⟳ of W evaluated at s=h+1; that is, M(h+1)=E(Wʰ). The inverse Mellin transform⟳ gives the density function of W.”

“In Chapters 2 and 7, the Mellin transform⟳ of the exponential function and the inverse Mellin transform⟳ of the Gamma function play⟳ key roles in demonstrating the equivalence of the modular relation and the functional equation. In proving the identities in this chapter, Mellin transforms also play⟳ central roles.”

CEFR level