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

Useful for solving various integral equations and PDEs. Fourier transforms, Laplace transforms, Mellin transforms, Hankel transforms…

Free resource: The transforms and applications handbook / edited by Alexander D. Poularikas. (Poularikas 2000)

## Fourier transform

See Fourier transforms.

## Laplace transform

TBD.

## Mellin transform

A transform with “Scale invariance like the Fourier transform has shift invariance”. Sounds fun.

## References

Adams, David R., and Lars I. Hedberg. 1999.

*Function Spaces and Potential Theory*. Springer Science & Business Media.Bruda, Glenn. 2022. “Maclaurin Integration: A Weapon Against Infamous Integrals.” arXiv.

Brychkov, I︠U︡ A., O. I. Marichev, and Nikolay V. Savischenko. 2019.

*Handbook of Mellin Tranforms*. Advances in Applied Mathematics. Boca Raton: CRC Press, Taylor & Francis Group.Davies, Brian. 2002.

*Integral Transforms and Their Applications*. 3rd edition. New York: Springer.Debnath, Lokenath, and Dambaru Bhatta. 2014.

*Integral Transforms and Their Applications*. 3rd edition. Chapman and Hall/CRC.Polyanin, A. D., and A. V. Manzhirov. 1998.

*Handbook of Integral Equations*. Boca Raton, Fla: CRC Press.Poularikas, Alexander D., ed. 2000.

*The Transforms and Applications Handbook*. 2nd ed. The Electrical Engineering Handbook Series. Boca Raton, Fla: CRC Press.Schiff, Joel L. 1999.

*The Laplace Transform: Theory and Applications*. 1999th edition. New York: Springer.Simon, Barry. 2015.

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