Expansion of free space Green function in Bessel function

[shaun_chou]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

 

[Jhero]

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.