Classical Electrodynamics · Problem 3.11

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

Physics — E&M · bessel-functions-fourier-series

Problem

A modified Bessel–Fourier series on the interval 0≤ρ≤a0 \leq \rho \leq a for an arbitrary function f(ρ)f(\rho) can be based on the ``homogeneous'' boundary conditions: At ρ=0\rho = 0, ρJν(kρ)dJν(k′ρ)dρ=0\quad \rho J_\nu(k\rho) \dfrac{dJ_\nu(k'\rho)}{d\rho} = 0 At ρ=a\rho = a, ddρln⁡[Jν(kρ)]=−λa\quad \dfrac{d}{d\rho} \ln[J_\nu(k\rho)] = -\dfrac{\lambda}{a} (λ\lambda real) The first condition restricts ν\nu. The second condition yields eigenvalues k=yνn/ak = y_{\nu n}/a, where yνny_{\nu n} is the nnth positive root of x dJν(x)/dx+λJν(x)=0x\, dJ_\nu(x)/dx + \lambda J_\nu(x) = 0. (a) Show that the Bessel functions of different eigenvalues are orthogonal in the usual way. (b) Find the normalization integral and show that an arbitrary function f(ρ)f(\rho) can be expanded on the interval in the modified Bessel–Fourier series f(ρ)=∑n=1∞AnJν(yνnρa)f(\rho) = \sum_{n=1}^{\infty} A_n J_\nu\left(\frac{y_{\nu n}\rho}{a}\right) with the coefficients AnA_n given by An=2a2[(1−ν2yνn2)Jν2(yνn)+(dJν(yνn)dyνn)2]−1∫0af(ρ) ρ Jν(yνnρa) dρA_n = \frac{2}{a^2}\left[\left(1 - \frac{\nu^2}{y_{\nu n}^2}\right)J_\nu^2(y_{\nu n}) + \left(\frac{dJ_\nu(y_{\nu n})}{dy_{\nu n}}\right)^2\right]^{-1} \int_0^a f(\rho)\, \rho\, J_\nu\left(\frac{y_{\nu n}\rho}{a}\right)\, d\rho

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