Complex integral representation of Dirac delta function?

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Discussion Overview

The discussion revolves around the complex integral representation of the Dirac delta function, specifically examining the validity of the expression \(\int e^{z^*(z-w)}dz^*\propto\delta(z-w)\). Participants explore the implications of this representation in a physics context, noting the lack of a specified domain of integration.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant states a known integral representation of the Dirac delta function and questions the validity of a complex integral representation related to it.
  • Another participant recalls a previous discussion on the same topic, indicating a potential lack of consensus or clarity on the matter.
  • A third participant shares a link to an article that may provide additional insights into the topic, suggesting it could be helpful for understanding the complex delta function.
  • A later reply expresses appreciation for the shared resource, indicating it may be a valuable starting point for further exploration.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the validity of the complex integral representation of the Dirac delta function, and multiple views on the topic remain present.

Contextual Notes

The discussion highlights the ambiguity regarding the domain of integration in the context of the proposed integral representation.

pellman
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We all know that [tex]\frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x')[/tex].

i am working a problem which appears to depend on the statement

[tex]\int e^{z^*(z-w)}dz^*\propto\delta(z-w)[/tex]

Does anyone know if this is valid?

[tex]\delta(z-w)[/tex] is defined in the usual way so that

[tex]\int{\delta(z-w)f(z)dz}=f(w)[/tex]

This is a physics problem, though, so the domain of integration is not specified and not clear to me.
 
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Hi, pellman

I just saw your thread and remembered reading a similar article on the net. Here it is, I guess it may be helpful:

http://homepages.physik.uni-muenchen.de/~Winitzki/no_distrib_limit.pdf
 
Last edited by a moderator:
That paper is a very good jumping off point. Thank you!
 

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