Complex integral representation of Dirac delta function?

In summary, the conversation discusses the validity of the statement that the integral of e^{z^*(z-w)}dz^* is proportional to the Dirac delta function, and whether it holds in a physics context. The domain of integration is not specified. One participant suggests a helpful article on the topic.
  • #1
pellman
684
5
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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  • #2
  • #3
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
 
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  • #4
That paper is a very good jumping off point. Thank you!
 

Related to Complex integral representation of Dirac delta function?

1. What is the complex integral representation of the Dirac delta function?

The complex integral representation of the Dirac delta function is a mathematical tool used in complex analysis and signal processing to represent the Dirac delta function as a complex integral. It is defined as δ(x) = (1/2π) * ∫-∞∞ eixt dt.

2. How is the complex integral representation of the Dirac delta function derived?

The complex integral representation of the Dirac delta function can be derived using the Fourier transform of the Dirac delta function and the inverse Fourier transform. By taking the Fourier transform of the Dirac delta function, we get δ(x) = 1/(2π) * ∫-∞∞ eixt dt. Then, by taking the inverse Fourier transform, we get the complex integral representation.

3. What are the advantages of using the complex integral representation of the Dirac delta function?

The complex integral representation of the Dirac delta function allows for easier manipulation and calculation of integrals involving the Dirac delta function compared to using the traditional definition. It also extends the use of the Dirac delta function to complex analysis and signal processing problems.

4. Can the complex integral representation of the Dirac delta function be used in higher dimensions?

Yes, the complex integral representation of the Dirac delta function can be extended to higher dimensions by using the multidimensional Fourier transform and its inverse. The resulting representation would be δn(x) = (1/(2π)n) * ∫Rn eixt dt, where n is the number of dimensions.

5. How is the complex integral representation of the Dirac delta function used in practical applications?

The complex integral representation of the Dirac delta function is used in various applications such as image processing, signal filtering, and solving differential equations. It allows for the simplification of complex mathematical expressions and provides a more intuitive understanding of the behavior of the Dirac delta function in these applications.

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