Mellin transform

The Mellin transform of a function $f$ is another function $φ$ such that

f (x) = \frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} x^{- s} φ (s) d s .

For a value $s$ , the output $φ (s)$ represents "how important" is the power $x^{- s}$ to construct, by "continuous linear combination" (i.e., by integration) the function $f$ . This coefficients can be calculated with

φ (s) = \int_{0}^{\infty} x^{s - 1} f (x) d x .

Seen as a kind of Laplace package

The Mellin transform is the same basis-change picture as the Laplace transform, applied after a linear map. See Fourier, Laplace and Dirichlet relationship#7. The Mellin Transform Same Package, Multiplicative Coordinates.

In this sense, Riemann zeta function $ζ (s)$ is the Mellin transform of the unit Dirac comb at the integers $\sum_{n = 1}^{\infty} δ (x - n)$ (see Riemann zeta function), and $Γ (s) = M {e^{- x}}$ .