Timeline for Exponential derivative of delta distribution?
Current License: CC BY-SA 3.0
9 events
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| Oct 27, 2017 at 15:17 | vote | accept | Kagaratsch | ||
| Oct 27, 2017 at 0:59 | comment | added | reuns | To make sense to $\delta(x-a)$ for $a \in \mathbb{C}$ you need to look at analytic functionals (the equivalent of distributions but acting on some space of analytic functions instead of the Schwartz space and $C^\infty_c$). In that case the Fourier/Laplace transform formulation of $\delta(x-a)$ makes sense on the smaller space of analytic functions being the Fourier/Laplace transform of a function decreasing faster than every exponential (a space containing the Gaussians) @Kagaratsch | |
| Oct 26, 2017 at 21:32 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Oct 26, 2017 at 21:14 | comment | added | Carlo Beenakker | Thank you all for your feedback. I have added a few references that may provide some justification for this delta function of a complex argument. The previous MO question @Nemo referred to was in response to a query what Paul Dirac meant when he wrote about the delta function of a "c-number". In that context the "c-number" is still purely real ("c" then stands for "classical", as opposed to "quantum"). So that is not directly related to this question. | |
| Oct 26, 2017 at 21:10 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Oct 26, 2017 at 15:25 | comment | added | Kagaratsch | @Nemo Very useful link, thank you! There Carlo Beenakker writes "Dirac never considered the delta function of a complex argument, only of real numbers." and suggests to split the delta function into two, one for real and one for imaginary part of the argument. In case at hand that would demand $n=0$. However, the expression $e^{-in\frac{\partial}{\partial x}}\delta(x)$ seems to allow for more general $n$ under integration against a test function. Thus my whole question above. | |
| Oct 26, 2017 at 15:21 | comment | added | Nemo | @CarloBeenakker correct me if I'm wrong, it seems you already have given negative answer to this question here mathoverflow.net/questions/118101/… ? | |
| Oct 26, 2017 at 15:15 | comment | added | Kagaratsch | I was under the impression that the integral representation of the delta function is only defined for real arguments $x$. That is why I'm not sure $\delta(x-in)$ exists? Which limit or integral representation does it have to come from, so that it converges and we get a single spike at a point $x-in=0$ in the complex plane? Perhaps you know of some reference that discusses this? | |
| Oct 26, 2017 at 15:07 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |