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Was there ever proposed a theory where $\delta(0)$ has a meaningful value or used in a formal way outside integrals?

Particularly, following Fourier transforms, we can formally obtain

$$\pi\delta(0)=\int_0^\infty dx=\int_{0^+}^\infty \frac1{x^2}dx$$

Were there attempts to treat these integrals as some quantities?

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  • $\begingroup$ Unless a surreal theory of integration I'm not aware of had been developped, I strongly doubt it. $\endgroup$ Jul 8 '17 at 9:57
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    $\begingroup$ You may want to have a look at Colombeau Algebras. $\endgroup$ Jul 8 '17 at 12:25
  • $\begingroup$ @Johannes Hahn I am currently reading Yegorov's book on Colomau algebra. But it seems it is not exactly I was thinking about... Still reading. $\endgroup$
    – Anixx
    Jul 8 '17 at 12:30
  • $\begingroup$ @Johannes Hahn well, this paper esc.fnwi.uva.nl/thesis/centraal/files/f59708808604356.pdf asks what is $\delta(0)$ but the answer is simply "some generalized complex number". It would be interesting to read some peper on Colombeau algebras but with emphasis on the generalized complex numbers rather than generalized functions. $\endgroup$
    – Anixx
    Jul 8 '17 at 13:14
  • $\begingroup$ It's interesting that this doesn't seem to work out either with NSA (because we can't explicitly define a particular hyperreal) or smooth infinitesimal analysis (because we don't have discontinuous functions). $\endgroup$ Jul 8 '17 at 19:27
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In physics this is a frequent device when we wish to quantize the electromagnetic field. We encounter Dirac delta functions $\delta(\omega-\omega')$ and need to make sense of the case $\omega=\omega'$. The way out is to imagine that the whole of space is enclosed in a cavity, with discrete frequencies $\omega_p=p\Delta$, $p=1,2,\ldots$, and then the Dirac delta becomes a Kronecker delta, $$\delta(\omega_p-\omega_p')=\Delta^{-1}\delta_{pp'},$$ so $\delta(0)=\Delta^{-1}$. At the end of the calculation we can then send the size of the cavity to infinity, so $\Delta\rightarrow 0$, and if we are calculating physically meaningful quantities this limit will be finite.

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    $\begingroup$ But is not this still used under integral? $\endgroup$
    – Anixx
    Jul 8 '17 at 12:31
  • $\begingroup$ no integrals over frequency are involved $\endgroup$ Jul 8 '17 at 12:38
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This works fine in Robinson's framework by choosing for example the Cauchy distribution (in the probability sense, not the Schwartz sense) with an infinitesimal value of the parameter. The "infinitely tall, infinitely narrow" delta-function this obtained returns the value (of a test function at the point when integrated against it) up to an infinitesimal. Robinson in his 1966 book proves the existence of delta functions which return the value of the test functions "on the nose" but these are a bit harder to define.

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