Classical Electrodynamics · Problem 3.16

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · bessel-functions-green-functions

Problem

(a) Starting from the Bessel differential equation and appropriate limiting procedures, verify the generalization of (3.108),

\frac{1}{k}\delta(k - k') = \int_0^{\infty} \rho J_\nu(k\rho) J_\nu(k'\rho)\, d\rho

or equivalently that

\frac{1}{\rho}\delta(\rho - \rho') = \int_0^{\infty} k J_\nu(k\rho) J_\nu(k\rho')\, dk

where

\text{Re}(\nu) > -1

. (b) Obtain the following expansion:

\frac{1}{|\mathbf{x} - \mathbf{x}'|} = \sum_{m=-\infty}^{\infty} \int_0^{\infty} dk\, e^{im(\phi-\phi')} J_m(k\rho) J_m(k\rho') e^{-k(z_> - z_<)}

\frac{1}{\sqrt{\rho^2 + z^2}} = \int_0^{\infty} e^{-k|z|} J_0(k\rho)\, dk

J_0\left(k\sqrt{\rho^2 + \rho'^2 - 2\rho\rho'\cos\phi}\right) = \sum_{m=-\infty}^{\infty} e^{im\phi} J_m(k\rho) J_m(k\rho')

e^{ik\rho\cos\phi} = \sum_{m=-\infty}^{\infty} i^m e^{im\phi} J_m(k\rho)

(d) From the last result obtain an integral representation of the Bessel function:

J_m(x) = \frac{1}{2\pi i^m} \int_0^{2\pi} e^{ix\cos\phi - im\phi}\, d\phi

Compare the standard integral representations.