Mellin transform

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English

English Wikipedia has an article on:

Mellin transform

Alternative forms

Mellin-transform (attributive use)

Etymology

Named after Finnish mathematician Hjalmar Mellin.

Noun

Mellin transform (plural Mellin transforms)

(mathematical analysis, number theory, statistics) An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.
- 1995, Lokenath Debnath, Integral Transforms and Their Applications, CRC Press, page 211:
  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,
  $M(h+1)=E(W^{h})$ .
  
  The inverse Mellin transform gives the density function of $W$ .
- 2008, Bruce C. Berndt, Marvin I. Knopp, Hecke's Theory of Modular Forms and Dirichlet Series, World Scientific, page 115:
  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.

Further reading

Mellin inversion theorem on Wikipedia.Wikipedia
Perron's formula on Wikipedia.Wikipedia
Asymptotic expansion on Wikipedia.Wikipedia
Fourier transform on Wikipedia.Wikipedia
Laplace transform on Wikipedia.Wikipedia
Dirichlet series on Wikipedia.Wikipedia
Riemann zeta function on Wikipedia.Wikipedia
Mellin transform on Encyclopedia of Mathematics
Mellin Transform on Wolfram MathWorld

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