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

“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

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

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*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.

*Handbook of Function and Generalized Function Transformations*. London: CRC Press.

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