Mellin transform
nounEtymology Named after Finnish mathematician Hjalmar Mellin.
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Mathematical analysis, Mathematics, Number theory, Sciences, Statistics An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.
- 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.1995, Lokenath Debnath, Integral Transforms and Their Applications, CRC Press, page 211:
- 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.
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- 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.2008, Bruce C. Berndt, Marvin I. Knopp, Hecke's Theory of Modular Forms and Dirichlet Series, World Scientific, page 115: