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 .

For example, $φ (- 1)$ should be the first derivative of $f$ evaluated at 0. But

φ (- 1) = \int_{0}^{\infty} x^{- 2} f (x) d x

is not $f^{'} (0)$ , or it is? Observe the similarity with Cauchy integral formula...

It is related to Fourier transform and therefore to Laplace transform.

It is somewhat related to Riemann zeta function.