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6 votes

Fourier cosine transform from Erdélyi's Tables of Integral Transforms

According to the Table Errata reported in Mathematics of Computation, vol. 65, no. 215, 1996, pp. 1379–1386, this entry in Erdélyi's Tables of Integral Transforms is flawed. The exponent in the denomi …
Michael Engelhardt's user avatar
43 votes

Why is the Fourier transform so ubiquitous?

From the point of view of physics, Fourier transforms are ubiquitous because they are expansions in eigenfunctions of the derivative operator - and the derivative operator is fundamental in many aspec …
Michael Engelhardt's user avatar
17 votes

Motivation and physical interpretation of the Laplace transform

The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\varepsilon )$ (a non-statistical concept) into the ca …
Michael Engelhardt's user avatar
1 vote

A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}...

You can of course write $$ k(x,t) = \exp \left[ e^{-t^2 /2} p\left( -\frac{d}{dt} \right) e^{t^2 /2} \right] \delta (x-t) $$ Not clear whether that affords you any sort of simplification you may be po …
Michael Engelhardt's user avatar
2 votes

Limit of an integral vs limit of the integrand

It is not unreasonable that a massless field behaves in a way that is totally different from a massive one with arbitrarily small mass. Already at an elementary level, you can always perform a Lorentz …
Michael Engelhardt's user avatar