Mellin transform

• Mellin-transform (attributive use)

Named after Finnish mathematician Hjalmar Mellin.

Mellin transform (plural Mellin transforms)

1. (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 ${\displaystyle X}$ is a positive random variable with density function ${\displaystyle f(x)}$, the Mellin transform ${\displaystyle M(s)}$ gives the ${\displaystyle (s-l)}$th moment of ${\displaystyle X}$. Hence Theorem 8.2.6 gives the Mellin transform of ${\displaystyle W}$ evaluated at ${\displaystyle s=h+1}$; that is,

     ${\displaystyle M(h+1)=E(W^{h})}$.

     The inverse Mellin transform gives the density function of ${\displaystyle W}$.

   • 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.

• 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