Contour integral with delta function

In summary, the conversation discusses using Cauchy's integral theorem to compute the integral of dz D^r delta(z) z^-m over a closed contour in the complex plane. It is noted that the function delta(z) is not strictly analytic, and there is a pole of order 'm' present. The conversation also mentions the possibility of finding the poles and recalling Cauchy's integral theorem, as well as the idea of using a sequence of functions to converge to the delta distribution. There is also discussion about the integration path and its intersection with the origin, as well as the meaning of D^r.
  • #1
mhill
189
1
Using Cauchy's integral theorem how could we compute

[tex] \oint _{C}dz D^{r} \delta (z) z^{-m} [/tex]

since delta (z) is not strictly an analytic function and we have a pole of order 'm' here C is a closed contour in complex plane
 
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  • #2
Can you find out for which z there is a pole?

If so, can you recall what Cauchy's integral theorem states?
 
  • #3
When you write 'isn't strictly analytic' you are implying (to me at least) it is really close to being one. It isn't even a function of a complex variable.

Should you want to do this for any reason, then why not try the normal limit via a sequence of functions that converge to the delta distribution?
 
  • #4
What kind of path is the C. Does it go through origo? If the integration path does not intersect the origo, isn't the integral zero because integrand is zero along the path? If the integration path intersects the origo, isn't it a divergent integral then?

What does [tex]D^r[/tex] mean? Is it a constant, or a derivative operator?
 

FAQ: Contour integral with delta function

What is a contour integral?

A contour integral is a type of integral that is computed along a path in the complex plane. It is used to evaluate functions that are not defined on the real numbers, or to calculate certain physical quantities in quantum mechanics and other fields.

What is the delta function?

The delta function, denoted by δ(x), is a mathematical function that is zero everywhere except at x=0, where it is infinite. It is often used in physics and engineering to represent a point source or a localized distribution of mass, charge, or energy.

How is the contour integral with delta function calculated?

The contour integral with delta function is calculated by first choosing a contour (path) in the complex plane and then integrating the product of the function being evaluated and the delta function along that contour. The integral is evaluated using the residue theorem, which involves finding the singularities of the integrand and the corresponding residues.

What is the physical significance of the contour integral with delta function?

The contour integral with delta function has various physical interpretations, depending on the context in which it is used. In quantum mechanics, it can be used to calculate transition amplitudes or to represent the wave function of a particle. In electromagnetism, it can be used to calculate the potential of a point charge or the electric field due to a charged line or surface. In fluid mechanics, it can be used to calculate the velocity field around a point vortex or a dipole.

Are there any applications of the contour integral with delta function in real-world problems?

Yes, the contour integral with delta function has numerous applications in various fields such as quantum mechanics, electromagnetism, fluid mechanics, and signal processing. It is used to solve problems involving point sources or localized distributions, and it is also used in the study of path integrals and Green's functions. Additionally, it has applications in the analysis of time series data and in the design of filters and other signal processing algorithms.

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