Dirac delta function on the complex plane?

In summary, the conversation discusses the proportionality between two integrals involving complex functions and the use of the Dirac delta function. The question of whether this proportionality is true is raised, and the possibility of separating the integrals into independent parts is also considered. The role of the conjugation function in the equation is also mentioned.
  • #1
pellman
684
5
Supposedly,

∫ ez*(z - z0)f(z) dz*dz

is proportional to f(z0) much in the same way that

(1/2π)∫ eiy(x - x0)f(x) dxdy
= ∫ δ(x - x0)f(x) dx
= f(x0)


Is this true? Could someone help convince me of it, or point me to a text?

I would say that even if true, it would be incorrect to say that

∫ ez*(z - z0)dz* = δ(z - z0)

because the integration over dz and dz* cannot be done independently in the same way that a surface integral over dxdy in the plane can (sometimes) be separated into independent integrations over x and y. Or can it?
 
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  • #2
##z^*## is only a function in ##z##, too. So what you have is ##\exp(F(z))## with ##F(z)=G(z)(z-z_0)## where ##G(z)## is the conjugation.

(I used capital letters in order to avoid confusion with your function ##f##.)
 

1. What is the Dirac delta function on the complex plane?

The Dirac delta function on the complex plane, also known as the complex delta function, is a mathematical function that is defined on the complex plane and has the properties of a generalized function. It is used to represent a point mass or impulse at the origin of the complex plane.

2. How is the Dirac delta function on the complex plane different from the one-dimensional delta function?

The Dirac delta function on the complex plane is a two-dimensional function, while the one-dimensional delta function is a one-dimensional function. Additionally, the complex delta function has complex-valued arguments and can be defined in polar coordinates, while the one-dimensional delta function only has real-valued arguments.

3. What are the properties of the Dirac delta function on the complex plane?

The Dirac delta function on the complex plane has similar properties to the one-dimensional delta function, such as being infinitely tall and narrow at the origin, and having an integral of one over the entire complex plane. However, it also has some unique properties, such as being circularly symmetric and having a value of zero at all points except for the origin.

4. How is the Dirac delta function on the complex plane used in mathematics and physics?

The Dirac delta function on the complex plane is used to model point sources in two-dimensional space, such as electric charges or vortices in fluid dynamics. It is also used in the theory of distributions to solve partial differential equations and in quantum mechanics to represent wavefunctions of particles.

5. Can the Dirac delta function on the complex plane be expressed in terms of other functions?

Yes, the Dirac delta function on the complex plane can be expressed in terms of other functions, such as the complex Gaussian function or the complex exponential function. However, these representations are not unique and depend on the chosen definition of the complex delta function.

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