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May 11

Dirac delta function

The Dirac delta is a notorious real-valued "function" that is infinite at x=0 and zero everywhere else. In real analysis it is treated as a generalized function (Schwartz distribution). Disclosure, I don't know what those really are, but their construction involves bump functions, which are continuously differentiable at all orders but are zero outside of a region.

In the complex plane of course, any continuously differentiable function is analytic so it must be either constant or unbounded, amirite? So there are no complex bump functions with those properties.

So, is there a complex version of the Dirac delta, and how is it mathematically "handled"? Thanks. 2602:243:2008:8BB0:F494:276C:D59A:C992 (talk) 00:03, 11 May 2024 (UTC)[reply]

The second illustration in the Dirac delta article shows it as the limit of sequence of zero-centered normal distributions, which do not have compact support; this works as well for most applications. So bump functions are not essential. Nevertheless, I don't think this will help in attempting to define a complex version. --Lambiam 06:41, 11 May 2024 (UTC)[reply]

Actually a more fundamental question: are Fourier series and Fourier transforms important in complex analysis? This is where the delta function comes up in the real case, more or less. 2602:243:2008:8BB0:F494:276C:D59A:C992 (talk) 08:02, 11 May 2024 (UTC)[reply]

The theory of Fourier series was developed well before Dirac came up with his delta function. It only plays a role in the theory of the Fourier transform for a purely periodic signal, not perturbed by any noise, something not found in actual practical applications. Even then, the delta function simplifies the presentation, but can be avoided using a mixed representation. I don't see how any of this can be generalized to deal with functions on the complex domain. --Lambiam 16:26, 11 May 2024 (UTC)[reply]

Fourier transforms are better defined on spaces of tempered distributions, proper subspaces of the spaces of distributions. The distribution spaces are dual to the Schwartz space of infinitely differentiable swiftly decreasing functions (the function and its derivatives decrease rapidly at infinity), instead of being dual to compactly supported C-infinity function spaces. And just as it can be considered as a limit of compactly supported C-inf bump functions, it can be considered as a limit of the normal distributions in the second illustration that Lambiam cites, which are tempered. The smaller here the space of test functions, the bigger the dual space. So measures - dual to spaces of continuous functions are a subset of distributions, the Dirac "function" being a measure. The degree k of a distribution corresponds to what dual Ck space it can be considered to come from.

The point of all this is that just as Ck functions are in C(k-1) down to C0 and all are in C-infinity spaces, compactly supported or tempered, one can consider the even smaller space of nicer functions, the analytic ones. And expect to get a bigger dual, distribution space. That leads to Sato's hyperfunctions which give an answer to the original question - "So, is there a complex version of the Dirac delta, and how is it mathematically "handled"". The idea is that one representation of hyperfunctions (on the real line) are as (differences of) boundary values of holomorphic functions on the upper & lower half-planes. Our article: "Informally, the hyperfunction is what the difference f-g would be at the real line itself." The difference of f and g is sort of converging to the hyperfunction as one gets closer to the real line. Which is how one avoids the fact that complex analysis prevents one from using bump functions or swiftly decreasing tempered functions as the OP notes.

But these games of course are only productive if the functions f or g are singular at the real line. So to finally get to the question, 1/2

π

iz has a simple pole singularity at 0, and that it - the function or its singularity - represents the Dirac function is precisely Cauchy's integral formula. Schwartz distributions can only have finite order - they can be thought of as nth order derivatives of functions which are not differentiable or even continuous. Sato Hyperfunctions can have infinite order. Functions with essential singularities, like e^(1/z) lead to such. Finally, as our article explains, one can consider the boundary value defined hyperfunctions as dual to the space of real analytic functions on the line. So that's the way of thinking complexly about Dirac delta. Maybe there are other ways, perhaps as yet undreamt.

Two relevant notes. At the first scientific conference I ever attended, I happened to sit next to Dirac himself. Same suit as in pictures of the Solvay conferences of the 20s. A venerable and awesome presence, beyond the perception of this bear of little brain. Second, why aren't or weren't hyperfunctions better known? Back in the 90s at a party where most were in our cups [me especially after a Red Army vet taught me to drink vodka like Red Army]. . . A Japanese mathematician there told me that it was an intended Japanese monopoly, for when books on Sato school microlocal, hyperfunction stuff were translated, pikchers and illustrations and examples were eliminated. This seemed to be true for one book I checked- the original was a lot thicker. I mentioned this to one of the very few non-Japanese experts - and he drily said the monopoly was pretty successful [back in those pre-Wikipedia etc days] - while the fellow I heard it from remembered nothing of our conversation when he had sobered up. :-) John Z (talk) 16:28, 19 May 2024 (UTC)[reply]