Expansion of free space Green function in Bessel function

In summary, the Green function in Bessel function is a mathematical function used to describe the propagation of waves in free space. It is important in physics and engineering, and is expanded using the method of separation of variables to approximate solutions to the Helmholtz equation. However, there are limitations to this expansion and alternative methods may need to be used in certain cases.
  • #1
shaun_chou
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Homework Statement


In Jackson 3.16 we have to prove the expansion [tex]\frac{1}{\left{|}\vec{x}-\vec{x'}\right{|}}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dke^{im(\phi-\phi')}J_m(k\rho)J_m(k\rho')e^{-k(z_{>}-z_{<})}[/tex]


Homework Equations





The Attempt at a Solution


I tried to use the techniques in the textbook but I only got [tex]G=\frac{1}{2}\sum e^{im(\phi-phi')}\int_0^{\infty}dke^{k(z-z'}N_m(k\rho_>)J_m(k\rho_<)[/tex]I just can't get the relationship of [tex]z_>[/tex] and [tex]z_<[/tex] in the expansion. Can anybody help me?
 
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  • #2


Thank you for your post. The expansion you are trying to prove is known as the multipole expansion, and it is a fundamental tool in electromagnetism. In order to understand the relationship between z_> and z_<, we need to first understand the context in which this expansion is used.

The multipole expansion is used to represent a general potential in terms of simpler, known potentials. In this case, we are trying to express the potential due to a point charge at position x' in terms of simpler terms. The terms in the expansion correspond to different types of charge distributions, such as a monopole, dipole, quadrupole, etc.

Now, let's look at the integral in the expression you have derived. The integral is over the magnitude of the wavevector k, and it is multiplied by a factor that depends on the angle between the two points, phi-phi'. This angle is related to the z-coordinates of the points, z_> and z_<, through the relation tan(phi-phi') = (z_>-z_<)/|x-x'|. This means that the difference between the z-coordinates of the two points is related to the angle between them. As m increases, the angle between the two points also increases, which means that the difference between the z-coordinates will also increase.

In summary, the z_> and z_< in the expansion represent the z-coordinates of the two points, and their difference is related to the angle between the points. I hope this helps clarify the relationship between these variables. Keep up the good work on your problem, and let us know if you have any further questions.
 

What is the Green function in Bessel function?

The Green function in Bessel function is a mathematical function used in quantum mechanics and electromagnetism to describe the propagation of waves in free space. It is a solution to the Helmholtz equation and is expressed in terms of Bessel functions.

Why is the Green function in Bessel function important?

The Green function in Bessel function is important because it allows us to solve problems involving wave propagation in free space, which is a fundamental concept in many areas of physics. It also has applications in engineering, such as in antenna design and signal processing.

How is the Green function in Bessel function expanded?

The Green function in Bessel function is typically expanded using the method of separation of variables. This involves expressing the function as a product of two simpler functions and then solving for each function separately. This results in a series of terms involving Bessel functions of different orders and arguments.

What is the significance of the expansion of the Green function in Bessel function?

The expansion of the Green function in Bessel function allows us to approximate the solution to the Helmholtz equation in free space. By truncating the series at a certain point, we can achieve a desired level of accuracy in our solution. This is particularly useful in practical applications where exact solutions may be difficult to obtain.

Are there any limitations to the expansion of the Green function in Bessel function?

Yes, there are limitations to the expansion of the Green function in Bessel function. The method of separation of variables may not always be applicable, and the series expansion may not converge for certain values of the arguments. In these cases, alternative methods or approximations may need to be used.

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