\[\renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\ff}[1]{\mathfrak{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\cc}[1]{\mathcal{#1}}\]

“An integral transform with scale invariance like the Fourier transform has shift invariance”. From my perspective, useful for analysing multiplicative products of random variables, their reciprocal and powers, much as the Fourier transform is useful for sums of scaled random variables.

For now, see the Mellin transform on Wikipedia.

Approachable references seem to be Bertrand, Bertrand, and Ovarlez (2000); Flajolet, Gourdon, and Dumas (1995); Galambos and Simonelli (2004).

## References

*Function Spaces and Potential Theory*. Springer Science & Business Media.

*The Transforms and Applications Handbook*, edited by Alexander D. Poularikas, 2nd ed. The Electrical Engineering Handbook Series. Boca Raton, Fla: CRC Press.

*Handbook of Mellin Tranforms*. Advances in Applied Mathematics. Boca Raton: CRC Press, Taylor & Francis Group.

*IEEE Transactions on Signal Processing*41 (12): 3275–92.

*Integral Transforms and Their Applications*. 3rd edition. New York: Springer.

*EURASIP J. Appl. Signal Process.*2007 (1): 75–75.

*Integral Transforms and Their Applications*. 3rd edition. Chapman and Hall/CRC.

*Advances in Applied Mathematics*20 (3): 285–99.

*Theoretical Computer Science*144 (1): 3–58.

*Products of random variables : applications to problems of physics and to arithmetical functions*. New York : Marcel Dekker.

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*Proceedings of the Physical Society. Section A*63 (10): 1101.

*arXiv:1905.10733 [Cs, Math, Stat]*, May.

*DAFX (Digital Audio Effects)*. Erlangen, Germany.

*Journal of Applied Probability*37 (2): 429–52.

*Handbook of Integral Equations*. Boca Raton, Fla: CRC Press.

*The Transforms and Applications Handbook*. 2nd ed. The Electrical Engineering Handbook Series. Boca Raton, Fla: CRC Press.

*The Laplace Transform: Theory and Applications*. 1999th edition. New York: Springer.

*Real Analysis*. A Comprehensive Course in Analysis 1.0. UNIVERSITIES PRESS.

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