What are the applications of Mellin transforms?

Question

I stumbled upon an integral transform called the Mellin transform, I found how it is defined but I don’t understand how this integral is supposed to converge, especially for positive values of n. Also, what are the applications of the Mellin transform, especially in pure maths ? And why is it defined this way, is it because of the gamma function ?

Answer 1

The Mellin transform is used to study functions that have power-law decay. It is defined as $$ \mathcal{M}(f)(s) = \int_0^\infty f(x)x^{s-1}\,dx \tag{1} $$ where $s=\sigma+it$ is a complex parameter. Its inverse is given by $$ f(x) = \frac{1}{2\pi i} \int_{\sigma_0-i\infty}^{\sigma_0+i\infty} \mathcal{M}(f)(s)x^{-s}\,ds \tag{2} $$ where $\sigma_0$ is a number such that the integral on the right of (2) converges. For example, suppose $f$ is defined by $f(x) = x^{-\alpha}$ with $\alpha > 0$. Then $$ \mathcal{M}(f)(s) = \int_0^\infty f(x)x^{s-1}\,dx = \int_0^\infty x^{-\alpha+s-1}\,dx = \Gamma(s-\alpha) $$ where we have used $\Gamma(t) = \int_0^\infty x^{t-1}e^{-x}\,dx$ for $t > 0$. Thus, the Mellin transform of $f$ is a meromorphic function with a pole of order $-\alpha$ at $s=\alpha$. Note that $f$ is defined for $x > 0$, so the integral on the left of (1) converges if $\mathrm{Re}(s) > \alpha$. The integral on the right of (2) converges if $\sigma_0 < \mathrm{Re}(s) < \alpha$. The Mellin transform has many other applications in pure and applied mathematics.

Answer 2

The Mellin transform is really "just" a different-coordinate version of Fourier transform, with coordinates changed by $x\to e^x$, mapping the real line "with addition" to the positive reals "with multiplication".

As with literal Fourier transform, the underlying point is that Mellin inversion (and/or Fourier inversion) expresses a given function as a superposition of eigenfunctions for the operator "differentiation" (or something close to it...)