Integral transforms
January 29, 2021 — January 30, 2021
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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)
1 Fourier transform
See Fourier transforms.
2 Laplace transform
Casual interpretation of the Laplace transform: Generalized Fourier transform. Both more powerful and more difficult, although the nature of the power and the difficulties depends upon where it is to be applied.
\[{\mathcal {L}}\{f\}(s)=\operatorname {E} \!\left[e^{-sX}\right]=\mathbb {E} \!\left[e^{-sX}\right] \]
3 Mellin transform
A transform with “Scale invariance like the Fourier transform has shift invariance”. Sounds fun.
4 Hankel transform
TBD. Covered implicitly in rotational symmetries.