Multipole Radiation Field

The field is a superposition of multipole field amplitudes. The first few terms of this superposition are

$$\psi_F(\xi,\tau,r,\theta) = \psi_0(\xi,\tau,r)$$

$$- e^{i\theta} \frac{\partial}{\partial r}\psi_1(\xi,\tau,r)$$

$$+ e^{2i\theta}\left( \frac{\partial^2}{\partial r^2} -\frac{1}{r}\frac{\partial}{\partial r} \right)\psi_2(\xi,\tau,r)$$

$$- e^{3i\theta}\left( \frac{\partial^3}{\partial r^3} -\frac{3}{r}\f... ...rtial r^2} +\frac{3}{r^2} \frac{\partial}{\partial r} \right)\psi_3(\xi,\tau,r)$$

$$+ e^{4i\theta}\left( \frac{\partial}{\partial r^4} -\frac{6}{r}\fra... ...ac{15}{r^3} \frac{\partial}{\partial r} \right)\psi_4(\xi,\tau,r)~+\quad \cdots$$

$$+ (\textrm{complex~conjugate~terms}~ \textrm{corresponding~to}~m=-1,-2,-3,\cdots)$$ (52)

whose explicit form is

$$\psi_F(\xi,\tau,r,\theta) = \int_0^\infty \left[ \frac{2S^0_I(\tau+\sinh^{-1}u,\xi')-2S^0_{II... ...inh^{-1}u,\xi')}{\sqrt{(\xi^2-\xi'^2-r^2)^2+(2\xi\xi')^2 }} \right]\xi'\, d\xi'$$

$$- e^{i\theta} \frac{\partial}{\partial r} \left[ \frac{2S^1_I(\tau+... ...u-\sinh^{-1}u)}{\sqrt{(\xi^2-\xi'^2-r^2)^2+(2\xi\xi')^2 }} \right]\xi' \, d\xi'$$

$$+ e^{2i\theta}\left( \frac{\partial^2}{\partial r^2} -\frac{1}{r}\f... ...h^{-1}u,\xi')}{\sqrt{(\xi^2-\xi'^2-r^2)^2+(2\xi\xi')^2 }}- \right]\xi' \, d\xi'$$

$$+ \quad \textrm{etc.}$$ (53)

where

$$u=\frac{\xi^2-\xi'^2-r^2}{2\xi\xi'}~.$$

It is evident that each multipole term has its own distinguishing angular ($\theta$) and radial ($r$) dependence.