In the theory of distributions, which gives rigorous meaning to statements like $$\int_{-\infty}^{+\infty} f(x) \delta(x) \,dx = f(0),$$ is there also a rigorous notion of, what the physicists would call the radial delta function, with the property $$\int_{0}^{\infty} f(x) \delta_r(x) dx = f(0)?$$ If so, what are the properties of $\delta_r(x)$? What are the suitable test functions $f(x)$?

I would also appreciate some references for further reading.

  • $\begingroup$ Ok so the question is really about that (the fact that the point is at the boundary), not the expression for dirac delta in different coordinates? Then I suggest you rephrase the question to be about that. As written it's easy to get the impression you are asking about the latter. $\endgroup$ – Winther Jun 8 '18 at 20:42
  • $\begingroup$ Similar question asked before (but no answers given) How does the Dirac delta function operate when its peak is at the boundary of an integral? See some of the comments though. $\endgroup$ – Winther Jun 8 '18 at 20:44
  • $\begingroup$ A while a go, one of our profs defined it as follows: Consider $f_\varepsilon(x)$ Which has the values of $1/\varepsilon$ for $|x|<\varepsilon/2$ and $0$ everywhere else, $\forall \varepsilon >0$. Now in terms of $f_\varepsilon$ we define $\delta(x)=\lim_{\varepsilon\rightarrow 0}f_\varepsilon$. With some tricks from analysis you can derive the formula for the integral. If I remember correctly, there may have also been a way to use Cauchy's integral formula and evaluating real integrals in the complex plane to get a sensible result but I am not very familiar with that approach $\endgroup$ – Jepsilon Jun 8 '18 at 21:01
  • $\begingroup$ What dimensionality of radial delta function are you interested in: 1, 2, or 3? Your integral explicitly asks about the 1-dmensional version, but the reference to physicists might imply the 2-dimensionalor or 3-dimensional version. Also, are you interested with the delta function only at the origin, or at other places in the n-dimensional space? Bracewell's book The Fourier Transform and Its Applications has a very accessible chapter 5 "The Impulse Symbol" that also gives a short treatment on 2-D and 3-D deltas and radial versions at the origin. $\endgroup$ – Andy Walls Jun 9 '18 at 12:35
  • $\begingroup$ @AndyWalls: the radial delta function is independent of the dimensionality, because I do not care about contributions from the Jacobian. I am just looking for properties of the delta function that returns the value of the test function at the boundary. $\endgroup$ – Fizikus Jun 9 '18 at 20:20

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