Some arguments for the wave equation in quantum theory 3

Proof. For the first part, suppose there exists \(\Psi\in S(T)\) satisfying the hypotheses, then taking Fourier coefficients of the equation, for \(m\in\mathcal{Z}\), and using integration by parts, we have that; \[\mathcal{F}\left({\partial^{2}\Psi\over \partial t^{2}}-{\partial^{2}\Psi\over \partial x^{2}}\right)(m)={d^{2}\mathcal{F}(\Psi)(m,t)\over dt^{2}}+m^{2}\mathcal{F}(\Psi)(m,t)=0\,.\] Solving the resulting ODE’s for \(m\neq 0\), we have that; \[\begin{cases}\mathcal{F}(\Psi)(m,t)=A_{m}e^{imt}+B_{m}e^{-imt}=(A_{m}+B_{m})\cos(mt)+i(A_{m}-B_{m})\sin(mt),\\ \mathcal{F}(\Psi)(0,t)=A+Bt,\end{cases}\]

where \((A_{m}+B_{m})=\mathcal{F}(\Psi)(m,0)=\mathcal{F}(\Psi_{0})(m),(imA_{m}-imB_{m})=\mathcal{F}({\partial \Psi\over \partial t})(m,0)=\mathcal{F}(\Psi_{1})(m),A=\mathcal{F}(\Psi_{0})(0),B=\mathcal{F}(\Psi_{1})(0)\). It follows that;

By the inversion theorem, \(\Psi\) is unique, and, using the fact that \(\Psi\) is real, symmetry properties of the Fourier coefficients \(\{a_{m},b_{m},a'_{m},b'_{m}\}\) and (2); \[\begin{aligned} \Psi(x,t)&=\sum_{m\in\mathcal{Z}}\mathcal{F}(\Psi)(m,t)e^{ixm}\\ &=\mathcal{F}(\Psi_{0})(0)+\mathcal{F}(\Psi_{1})(0)t+\sum_{m\in\mathcal{Z}_{\neq 0}}(\mathcal{F}(\Psi)(m,0)\cos(mt)\\ &+{\mathcal{F}(\Psi_{1})(m)\over m}\sin(mt))(\cos(mx)+i\sin(mx))\\ &=a_{0}+a'_{0}t+\sum_{m\in\mathcal{Z}_{\neq 0}}((a_{m}-ib_{m})\cos(mt)(\cos(mx)+i\sin(mx))\\ &+\sum_{m\in\mathcal{Z}_{\neq 0}}\left({a'_{m}-ib'_{m}\over m}\right)\sin(mt))(\cos(mx)+i\sin(mx))\\ &=a_{0}+a'_{0}t+\sum_{m\in\mathcal{Z}_{\neq 0}}a_{m}\cos(mx)\cos(mt)+\sum_{m\in\mathcal{Z}_{\neq 0}}b_{m}\sin(mx)\cos(mt)\\ &+\sum_{m\in\mathcal{Z}_{\neq 0}}{a'_{m}\over m}\cos(mx)\sin(mt)+\sum_{m\in\mathcal{Z}_{\neq 0}}{b'_{m}\over m}\sin(mx)\sin(mt) \end{aligned}\] \[\begin{aligned} &=a_{0}+a'_{0}t+2\sum_{m\in\mathcal{Z}_{>0}}a_{m}\cos(mx)\cos(mt)+2\sum_{m\in\mathcal{Z}_{>0}}b_{m}\sin(mx)\cos(mt)\\ &+2\sum_{m\in\mathcal{Z}_{>0}}{a'_{m}\over m}\cos(mx)\sin(mt)+2\sum_{m\in\mathcal{Z}_{>0}}{b'_{m}\over m}\sin(mx)\sin(mt)\\ \end{aligned}\] as required. It is easily checked that the above series also defines \(\Psi\in S(T)\) with the required properties, settling the existence question.

For the second part, \(\{\rho,J\}\) satisfy the continuity equation, by the definition of \(J\) and the fundamental theorem of calculus. By inspection of the series for \(\Psi\), it is clear that \(J\in S(T)\). Moreover; \[P'(t)=\int\limits_{-\pi}^{\pi}{\partial \rho\over \partial t}(x,t)dx=\int\limits_{-\pi}^{\pi}-{\partial J\over \partial x}(x,t)dx=J(-\pi)-J(\pi)=0\,,\] so that \(P(t)=P(0)\) is constant. Differentiating under the integral sign, using the fact that \(\rho\) satisfies the wave equation, and using the fundamental theorem of calculus again, we have; \[{\partial J\over \partial t}=\int\limits_{-\pi}^{x}-{\partial^{2}\rho\over \partial t^{2}}dt=\int\limits_{-\pi}^{x}-{\partial^{2}\rho\over \partial x^{2}}dx=-{\partial\rho\over \partial x}+{\partial\rho\over \partial x}\Bigg|_{-\pi}\,.\] If \(\Psi_{0}\) and \(\Psi_{1}\) are symmetric, we have the above coefficients \(\{b_{m},b_{m}'\}\) are zero, and \({\partial\rho\over \partial x}\) expands as a series in \(\sin(mx)\). It follows that; \[{\partial\rho\over \partial x}\Bigg|_{-\pi}=0;\quad \text{and}\quad {\partial J\over \partial t}=-{\partial\rho\over \partial x}\,,\] so that, combined with the continuity equation, and the fact that the partial derivatives commutes, we obtain that; \[{\partial^{2} J\over \partial t^{2}}=-{\partial^{2}\rho\over \partial x\partial t}=-{\partial^{2}\rho\over \partial t\partial x}={\partial^{2} J\over \partial x^{2}}\,.\] Moreover, a simple calculation, using the definition of \(J\) shows that; \[\begin{aligned} \rho(x,t)&=a_{0}+a_{0}'t+2\sum_{m\in\mathcal{Z}>0}a_{m}\cos(mx)\cos(mt)+2\sum_{m\in\mathcal{Z}>0}{a'_{m}\over m}\cos(mx)\sin(mt),\\ J(x,t)&=-a'_{0}\pi-a'_{0}x+2\sum_{m\in\mathcal{Z}>0}a_{m}\sin(mx)\sin(mt)-2\sum_{m\in\mathcal{Z}>0}{a'_{m}\over m}\sin(mx)\cos(mt). \end{aligned}\] ◻

Proof. Using Lemma 1, it is sufficient to prove that, for \(-\pi\leq \theta<\pi\), \(t\in {\mathcal{R}}\); \[\ label{lem3.1*}div(\overline{J})|_{(1,\theta,0,t)}=J'(\theta,t). \tag{3}\] Omitting the \(t\) for ease of notation, and letting \(\overline{J}=(J())\), we have that; \[div(-J(\theta)\sin(\theta),J(\theta)\cos(\theta),0)={\partial \over \partial x}(-J(\theta)\sin(\theta))+{\partial \over \partial y}(J(\theta)\cos(\theta)).\] We have; \[\begin{aligned} {\partial \over \partial x}(-J(\theta)\sin(\theta))&=-\left({\partial J\over \partial x}\sin(\theta)+J(\theta){\partial \sin(\theta)\over \partial x}\right)\\ &=-\left(J'(\theta){\partial \theta\over \partial x}\sin(\theta)+J(\theta){\partial y\over \partial x}\right),\qquad \text{as}\; r=1\;\text{ and }\;\sin(\theta)={y\over r}=y\\ &=-\left(J'(\theta)\sin(\theta){\partial \theta\over \partial x}\right),\qquad \text{as}\; {\partial y\over \partial x}=0\\ &=-(J'(\theta)\sin(\theta)-y)=J'(\theta)\sin(\theta)y,\\&\qquad \text{as}\; \theta=\tan^{-1}\left({y\over x}\right)\;\text{ and }\;{\partial \theta\over \partial x}={{-y\over x^{2}}\over 1+({y\over x})^{2}}={-y\over x^{2}+y^{2}}=-y,\;\text{ with }\;r=1. \end{aligned}\] Similarly; \[\begin{aligned} {\partial \over \partial y}(J(\theta)\cos(\theta))&=\left({\partial J\over \partial y}\cos(\theta)+J(\theta){\partial \cos(\theta)\over \partial y}\right)\\ &=\left(J'(\theta){\partial \theta\over \partial y}\cos(\theta)+J(\theta){\partial x\over \partial y}\right),\qquad \text{as}\; r=1\;\text{ and }\;\cos(\theta)={x\over r}=x\\ &=\left(J'(\theta)\cos(\theta){\partial \theta\over \partial y}\right),\qquad \text{as}\; {\partial x\over \partial y}=0\\ &=\left(J'(\theta)\cos(\theta)x\right) =J'(\theta)\cos(\theta)x\\ &\qquad \text{as}\;\theta=tan^{-1}\left({y\over x}\right)\;\text{ and }{\partial \theta\over \partial y}={{1\over x}\over 1+({y\over x})^{2}}={x\over x^{2}+y^{2}}=x,\;\text{ with }\;r=1. \end{aligned}\] It follows that; \[div(-J(\theta)\sin(\theta),J(\theta)\cos(\theta),0)=J'(\theta)\sin(\theta)y+J'(\theta)\cos(\theta)x=J'(\theta)(\sin^{2}(\theta)+\cos^{2}(\theta))=J'(\theta),\] as \(x=r\cos(\theta)=\cos(\theta)\), \(x=r\sin(\theta)=\sin(\theta)\), when \(r=1\). ◻

Proof. We have that \({\partial \rho\over \partial t}=0\), and \[\begin{aligned} div(\overline{J})&={\partial (-w(r)\sin(\theta))\over \partial x}+{\partial (w(r)\cos(\theta))\over \partial y}\\ &=-{\partial w\over \partial x}\sin(\theta)-w{\partial \sin(\theta)\over \partial x}+{\partial w\over \partial y}\cos(\theta)+w{\partial \cos(\theta)\over \partial y}\\ &=-w'(r){\partial r\over \partial x}\sin(\theta)-w{\partial \sin(\theta)\over \partial x}+w'(r){\partial r\over \partial y}\cos(\theta)+w{\partial \cos(\theta)\over \partial y}\\ &=-w'(r){x\over r}\sin(\theta)-{w(r)(-\sin(\theta)\cos(\theta))\over r}+w'(r){y\over r}\cos(\theta)-w(r){(\sin(\theta)\cos(\theta))\over r}\\ &=-w'(r)\cos(\theta)\sin(\theta)-{w(r)(-\sin(\theta)\cos(\theta))\over r}+w'(r)\sin(\theta)\cos(\theta-w(r){(\sin(\theta)\cos(\theta))\over r}=0 \end{aligned}\] with \(x=r \cos(\theta)\), \(y=r \sin(\theta)\),(¹).

Conversely, suppose that \(div(\overline{J})=0\), with \(\overline{J}(r,\theta,t)=\rho(r,\theta)(-\sin(\theta),\cos(\theta))\). Then

\[\begin{aligned} &{\partial (-\rho(r,\theta) \sin(\theta)) \over \partial x}+{\partial (\rho(r,\theta) \cos(\theta)) \over \partial y}=-{\partial \rho\over \partial x}\sin(\theta)-\rho{\partial \sin(\theta)\over \partial x}+{\partial \rho\over \partial y}\cos(\theta)+\rho{\partial \cos(\theta)\over \partial y}\\ &\qquad=-\left({\partial \rho\over \partial r}{\partial r\over \partial x}+{\partial \rho\over \partial \theta}{\partial \theta\over \partial x}\right)\sin(\theta)-\rho\left({-\sin(\theta)\cos(\theta)\over r}\right)+\left({\partial \rho\over \partial r}{\partial r\over \partial y} +{\partial \rho\over \partial \theta}{\partial \theta\over \partial y}\right)\cos(\theta)+\rho\left({-\sin(\theta)\cos(\theta)\over r}\right), \end{aligned}\] using footnote 2, \({\partial \theta\over \partial x}={-\sin(\theta)\over r}\), \({\partial \theta\over \partial y}={\cos(\theta)\over r}\). Therefore

\[div(\overline{J})=\left({\partial \rho\over \partial r}{\partial r\over \partial y}+{\partial \rho\over \partial \theta}{\partial \theta\over \partial y}\right)\cos(\theta)-\left({\partial \rho\over \partial r}{\partial r\over \partial x}+{\partial \rho\over \partial \theta}{\partial \theta\over \partial x}\right)\sin(\theta).\] It follows that \[div(\overline{J})={\partial \rho\over \partial r}{y\over r}\cos(\theta)+{\partial \rho\over \partial \theta}{\partial \theta\over \partial y}\cos(\theta)-{\partial \rho\over \partial r}{x\over r}\sin(\theta)-{\partial \rho\over \partial \theta}{\partial \theta\over \partial x}\sin(\theta)\] with \(y=r\sin(\theta)\) and \(x=r\cos(\theta)\), using footnote 1 again; \({\partial r\over\partial x}={x\over r}=\cos(\theta)\), \({\partial r\over\partial y}={y\over r}=\sin(\theta)\). Hence \[div(\overline{J})=\left({\partial \rho\over \partial r}\sin(\theta)\cos(\theta)+{\partial \rho\over \partial \theta}{\cos^{2}(\theta)\over r}\right)-\left({\partial \rho\over \partial r}\sin(\theta)\cos(\theta)+{\partial \rho\over \partial \theta}{\sin^{2}(\theta)\over r}\right)={\partial \rho\over \partial \theta}{1\over r}=0.\] If \(r\neq 0\), \({\partial \rho\over \partial \theta}=0\), so \(\rho\) is independent of \(\theta\). ◻

Proof. By Lemma 3, any circular flow satisfying the continuity equation on \(Ann(1,\epsilon)\) has a density \(\rho\), depending only on \(r\). In particular, \(\rho|_{S(1)}\) is constant, contradicting the hypothesis. ◻

Proof. Suppose that \(\rho(\theta,r)\) is smooth and independent of time on the annulus, defined by \[Ann(1,\epsilon,\delta)=\{(\theta,r):-\pi\leq \theta<\pi, 1-\delta<r<1+\epsilon\},\] with smooth \(\overline{J}(\theta,r)\), satisfying the continuity equation, and \(\overline{J}(\theta,r)=(J_{1}(\theta,r),J_{2}(\theta,r))=w_{1}(\theta,r)\hat{\overline{r}}+w_{2}(\theta,r)\hat{\overline{\theta}}\), where \(\hat{\overline{r}}=(\cos(\theta),\sin(\theta))\) and \(\hat{\overline{\theta}}=(-\sin(\theta),\cos(\theta))\). It follows that \[\begin{aligned} \overline{J}(\theta,r)&=w_{1}(\theta,r)(\cos(\theta),\sin(\theta))+w_{2}(\theta,r)(-\sin(\theta),\cos(\theta))\\ &=(w_{1}\cos(\theta)-w_{2}\sin(\theta),w_{1}\sin(\theta)+w_{2}\cos(\theta)). \end{aligned}\] Using the hypotheses, that \(div(\overline{J})={\partial\rho\over \partial t}=0\), we have that \[\begin{aligned} {\partial J_{1}\over \partial x}&={\partial w_{1}\over \partial x}\cos(\theta)+w_{1}{\partial(\cos(\theta))\over \partial x}-{\partial w_{2}\over \partial x}\sin(\theta)-w_{2}{\partial(\sin(\theta))\over \partial x}\\ &={\partial w_{1}\over \partial x}\cos(\theta)+w_{1}-\sin(\theta){-y\over r^{2}}-{\partial w_{2}\over \partial x}\sin(\theta)-w_{2}\cos(\theta){-y\over r^{2}}\\ &={\partial w_{1}\over \partial x}\cos(\theta)+w_{1}{\sin(\theta)y\over r^{2}}-{\partial w_{2}\over \partial x}\sin(\theta)+w_{2}{\cos(\theta)y\over r^{2}}, \end{aligned}\] \[\begin{aligned} {\partial J_{2}\over \partial y}&={\partial w_{1}\over \partial y}\sin(\theta)+w_{1}{\partial(\sin(\theta))\over \partial y}+{\partial w_{2}\over \partial y}\cos(\theta)+w_{2}{\partial(\cos(\theta))\over \partial y}\\ &={\partial w_{1}\over \partial y}\sin(\theta)+w_{1}\cos(\theta){x\over r^{2}}+{\partial w_{2}\over \partial y}\cos(\theta)+w_{2}-\sin(\theta){x\over r^{2}}\\ &={\partial w_{1}\over \partial y}\sin(\theta)+w_{1}\cos(\theta){x\over r^{2}}+{\partial w_{2}\over \partial y}\cos(\theta)-w_{2}{\sin(\theta)x\over r^{2}}. \end{aligned}\] Therefore \[\label{lem5-eq1} div(\overline{J})=grad(w_{1})\centerdot \hat{\overline{r}}+{w_{1}\over r^{2}}(\overline{v_{\theta}}\centerdot flip(\overline{r}))+grad(w_{2})\centerdot \hat{\overline{\theta}}+{w_{2}\over r^{2}}(\overline{w_{\theta}}\centerdot flip(\overline{r})), \tag{4}\] where \(\overline{v_{\theta}}=(\sin(\theta),\cos(\theta))\), \(\overline{w_{\theta}}=(\cos(\theta),-\sin(\theta))\), and \(flip(\overline{r})=(y,x)\). We find \(\{\alpha,\beta\}\) such that \(\alpha\hat{\overline{\theta}}+\beta\hat{\overline{r}}=\overline{v_{\theta}}\). This is equivalent to finding \(\{\alpha,\beta\}\) such that \(M_{\theta}\overline{v_{\alpha,\beta}}=\overline{v_{\theta}}\), where \((M_{\theta})_{1,2}=(M_{\theta})_{2,1}=\cos(\theta)\) and \((M_{\theta})_{1,1}=-(M_{\theta})_{2,2}=-\sin(\theta)\) and \(\overline{v_{\alpha,\beta}}=(\alpha,\beta)\). We have, \[\overline{v_{\alpha,\beta}}=M_{\theta}^{-1}\overline{v_{\theta}}={1\over -\sin^{2}(\theta)-\cos^{2}(\theta)}N_{\theta}\overline{v_{\theta}}=-(\sin^{2}(\theta)-\cos^{2}(\theta),-2\sin(\theta)\cos(\theta))=(\cos(2\theta),\sin(2\theta)),\] where \((N_{\theta})_{1,2}=(N_{\theta})_{2,1}=-\cos(\theta)\) and \((N_{\theta})_{1,1}=-(N_{\theta})_{2,2}=\sin(\theta)\).

It follows that \(\alpha=\cos(2\theta)\), \(\beta=\sin(2\theta)\). Similarly, we find \(\{\alpha,\beta\}\) such that \(\alpha\hat{\overline{\theta}}+\beta\hat{\overline{r}}=\overline{w_{\theta}}\). Again, this is equivalent to finding \(\{\alpha,\beta\}\) such that \(M_{\theta}\overline{v_{\alpha,\beta}}=\overline{w_{\theta}}\). We have \[\overline{v_{\alpha,\beta}}=M_{\theta}^{-1}\overline{w_{\theta}}={1\over -\sin^{2}(\theta)-\cos^{2}(\theta)}N_{\theta}\overline{w_{\theta}}=-(2\sin(\theta)\cos(\theta),-\cos^{2}(\theta)+\sin^{2}(\theta))=(-\sin(2\theta),\cos(2\theta)).\] It follows that \(\alpha=\sin(2\theta)\), \(\beta=\cos(2\theta)\). Substituting in (4), we obtain \[\ label{lem5-eq2} div\left(\overline{J}\right) grad\left(w_{1}\right)\centerdot \hat{\overline{r}}+{w_{1}\over r^{2}}\left(\cos\left(2\theta\right)\hat{\overline{\theta}}+\sin\left(2\theta\right)\hat{\overline{r}}\right)\centerdot flip\left(\overline{r}\right)\nonumber\\ +grad\left(w_{2}\right)\centerdot \hat{\overline{\theta}}+{w_{2}\over r^{2}}\left(-\sin\left(2\theta\right)\overline{\theta}+\cos\left(2\theta\right){\overline{r}}\right)\centerdot flip\left(\overline{r}\right)\nonumber\\ \left(grad\left(w_{1}\right)+{w_{1}\over r^{2}}flip\left(\overline{r}\right)\sin\left(2\theta\right)+{w_{2}\over r^{2}}flip\left(\overline{r}\right)\cos\left(2\theta\right)\right)\centerdot \hat{\overline{r}}\nonumber\\ +\left(grad\left(w_{2}\right)-{w_{2}\over r^{2}}flip\left(\overline{r}\right)\sin\left(2\theta\right)+{w_{1}\over r^{2}}flip\left(\overline{r}\right)\cos\left(2\theta\right)\right)\centerdot \hat{\overline{\theta}}. \tag{5}\] We have that \(\overline{r}=r\hat{\overline{r}}\) and determine \(\{\alpha,\beta\}\) such that \(flip(\overline{r})=\alpha{\overline{\theta}}+\beta\hat{\overline{r}}\). Similarly to the above, we obtain \[\overline{v_{\alpha,\beta}}=-(r\sin^{2}(\theta)-r\cos^{2}(\theta),-2r\sin(\theta)\cos(\theta))=(r\cos(2\theta),r\sin(2\theta))=(r\cos(2\theta), r\sin(2\theta)),\] so that \(\alpha=r\cos(2\theta)\), \(\beta=r\sin(2\theta)\), and \(flip(\overline{r})=r\cos(2\theta)\hat{\overline{\theta}}+r\sin(2\theta)\hat{\overline{r}}\). Substituting into (5), and using the fact that \(\{\hat{\overline{r}},\hat{\overline{\theta}}\}\) are orthonormal, we obtain

\[\ label{lem5-eq3} div\left(\overline{J}\right) \left(grad\left(w_{1}\right)+\left({w_{1}\sin\left(2\theta\right)\over r^{2}}+{w_{2}\cos\left(2\theta\right)\over r^{2}}\right)\left(r\cos\left(2\theta\right)\hat{\overline{\theta}}+r\sin\left(2\theta\right){\overline{r}}\right)\right)\centerdot\hat{\overline{r}}\nonumber\\ +\left(grad\left(w_{2}\right)+\left({-w_{2}\sin\left(2\theta\right)\over r^{2}}+{w_{1}\cos\left(2\theta\right)\over r^{2}}\right)\left(r\cos\left(2\theta\right)\hat{\overline{\theta}}+r\sin\left(2\theta\right){\overline{r}}\right)\right)\centerdot\hat{\overline{\theta}},\nonumber\\ grad\left(w_{1}\right)\centerdot\hat{\overline{r}}+\left({\left(w_{1}\sin\left(2\theta\right)\right)r\sin\left(2\theta\right)\over r^{2}}+{\left(w_{2}\cos\left(2\theta\right)\right)r\sin\left(2\theta\right)\over r^{2}}\right)\nonumber\\ +grad\left(w_{2}\right)\centerdot\hat{\overline{\theta}}+\left({\left(w_{1}\cos\left(2\theta\right)\right)r\cos\left(2\theta\right)\over r^{2}}-{\left(w_{2}\sin\left(2\theta\right)\right)r\cos\left(2\theta\right)\over r^{2}}\right)\, \tag{6}\] Writing \(grad(w_{1})=\alpha\hat{\overline{r}}+\beta\hat{\overline{\theta}}\) and \(grad(w_{2})=\gamma\hat{\overline{r}}+\delta\hat{\overline{\theta}}\) with \(\alpha=grad(w_{1})\centerdot\hat{\overline{r}}\) and \(\delta=grad(w_{2})\centerdot\hat{\overline{\theta}}\), we obtain from (6) and \(div(\overline{J})=0\) that \[\label{lem5-eq4} \alpha+\delta={-w_{1}\over r}={-w_{1}\over r}. \tag{7}\] We have that \[\alpha=grad(w_{1})\centerdot\hat{\overline{r}}={\partial w_{1}\over \partial x}\cos(\theta)+{\partial w_{1}\over \partial y}\sin(\theta),\] \[\label{lem5-eq5} \delta=grad(w_{2})\centerdot\hat{\overline{\theta}}={\partial w_{2}\over \partial x}-\sin(\theta)+{\partial w_{2}\over \partial y}\cos(\theta). \tag{8}\] We have, using the calculations for \(\{{\partial \theta\over \partial x},{\partial\theta\over \partial y},{\partial r\over \partial x},{\partial r\over \partial y}\}\), from the previous lemma, and the chain rule; \[\label{lem5-eq6}\begin{cases} {\partial w_{1}\over \partial x}={\partial w_{1}\over \partial \theta}{-\sin(\theta)\over r}+{\partial w_{1}\over \partial r}\cos(\theta),\\ {\partial w_{1}\over \partial y}={\partial w_{1}\over \partial \theta}{\cos(\theta)\over r}+{\partial w_{1}\over \partial r}\sin(\theta),\\ {\partial w_{2}\over \partial x}={\partial w_{2}\over \partial \theta}{-\sin(\theta)\over r}+{\partial w_{2}\over \partial r}\cos(\theta),\\ {\partial w_{2}\over \partial y}={\partial w_{2}\over \partial \theta}{\cos(\theta)\over r}+{\partial w_{2}\over \partial r}\sin(\theta).\end{cases}. \tag{9}\] It follows from (8) and (9) that \[\begin{aligned} \alpha&=\left({\partial w_{1}\over \partial \theta}{-\sin(\theta)\over r}+{\partial w_{1}\over \partial r}\cos(\theta)\right)\cos(\theta)+\left({\partial w_{1}\over \partial \theta}{\cos(\theta)\over r}+{\partial w_{1}\over \partial r}\sin(\theta)\right)\sin(\theta),\\ \delta&=\left({\partial w_{2}\over \partial \theta}{-\sin(\theta)\over r}+{\partial w_{2}\over \partial r}\cos(\theta)\right)-\sin(\theta)+\left({\partial w_{2}\over \partial \theta}{\cos(\theta)\over r}+{\partial w_{2}\over \partial r}\sin(\theta)\right)\cos(\theta). \end{aligned}\] Simplifying and substituting into (7), we obtain \({\partial w_{1}\over \partial r}+{1\over r}{\partial w_{2}\over \partial \theta}={-w_{1}\over r}\) and rearranging \(r{\partial w_{1}\over \partial r}+w_{1}=-{\partial w_{2}\over \partial \theta}\). Fixing \(\theta_{0}\) and multiplying by \(p(\theta_{0},r)\), we obtain \[p(r,\theta_{0})r{dw_{1}^{\theta_{0}}\over dr}+p(r,\theta_{0})w_{1}^{\theta_{0}}=-p(r,\theta_{0}){\partial w_{2}\over \partial \theta}^{\theta_{0}}.\] Letting \[\label{lem5-eq7} s(r,\theta_{0})=p(r,\theta_{0})r, \tag{10}\] we have that \([s(r,\theta_{0})w_{1}(r,\theta_{0})]'=s'(r,\theta_{0})w_{1}+sw_{1}'(r,\theta_{0})\) so, equating coefficients, we require \[\label{lem5-eq8} s'(r,\theta_{0})=p(r,\theta_{0}). \tag{11}\] Using (10),(11), and, assuming \(p(r,\theta_{0})\neq 0\), we obtain \({s'(r,\theta_{0})\over s(r,\theta_{0})}={1\over r}\) \(ln(s(r,\theta_{0}))=ln(r)+d(\theta_{0})\) and, taking exponentials; \(s(r,\theta_{0})=A(\theta_{0})r\) where \(A(\theta_{0})=e^{d(\theta_{0})}\); \[[A(\theta_{0})r^{c(\theta_{0})}w_{1}(r,\theta_{0})]'=-p(r,\theta_{0}){\partial w_{2}\over \partial \theta}^{\theta_{0}},\] where \(p(r,\theta_{0})=A(\theta_{0})\). Integrating, and using the fundamental theorem of calculus, we obtain \[[A(\theta_{0})r w_{1}(r,\theta_{0})]_{1-\epsilon}^{r_{0}}=\int\limits_{1-\epsilon}^{r_{0}}-A(\theta_{0}){\partial w_{2}\over \partial \theta}^{\theta_{0}}dr.\] Case 1; For a given \(\epsilon,\delta>0\), and \(1-\epsilon<r_{0}<1+\delta\), we obtain \[r_{0} w_{1}(r_{0},\theta_{0})-(1-\epsilon) w_{1}(1-\epsilon,\theta_{0})=\int\limits_{1-\epsilon}^{r_{0}}-{\partial w_{2}\over \partial \theta}^{\theta_{0}}dr,\] so free to choose a smooth \(w_{2}\) on \(Ann(\epsilon,\delta,1)\) and a smooth boundary condition for \(w_{1}\) on \(S^{1}(1-\epsilon)\), to obtain \(w_{1}\).

Case 2; Letting \(\epsilon=1,\delta=0\), and, assuming \(w_{1}\) is defined at \((0,0)\), we must have that, for \(r_{0}>0\); \[r_{0} w_{1}(r_{0},\theta_{0})=\int\limits_{0}^{r_{0}}-{\partial w_{2}\over \partial \theta}^{\theta_{0}}dr.\] By L’Hopital’s rule, the fact that \(r(0)=0\), \(a(0)=0\), where \(a(r)=\int\limits_{0}^{r}-{\partial w_{2}\over \partial \theta}^{\theta_{0}}dr'\), and the Fundamental Theorem of Calculus, \(a'(0)=-{\partial w_{2}\over \partial \theta}(0,0)\), we have that \[\lim\limits_{r_{0}\rightarrow 0}{1\over r_{0}}\int\limits_{0}^{r_{0}}-{\partial w_{2}\over \partial \theta}^{\theta_{0}}dr={a'(0)\over r'(0)}=-{\partial w_{2}\over \partial \theta}^{\theta_{0}}(0),\] so defining \(w_{1}(0,0)=-w_{2}(0,0)\) generates a continuous solution, provided \(\lim\limits_{r\rightarrow 0}-{\partial w_{2}\over \partial \theta}^{\theta_{1}}=\lim\limits_{r\rightarrow 0}-{\partial w_{2}\over \partial \theta}^{\theta_{2}}\), for all \(\{\theta_{1},\theta_{2}\}\subset [-\pi,\pi)\), this is true if \(w_{2}\) is analytic in \((x,y)\),(²). ◻

Proof. Let \(\Phi_{\epsilon}(r)=e^{-{1\over 1-{(r-1)^{2}\over \epsilon^{2}}}}\), if \(r\in (1-\epsilon,1+\epsilon)\) and \(\Phi_{\epsilon}(r)=0\) otherwise, \(r>0\), then \(\Phi_{\epsilon}\) is smooth on \(\mathcal{R}_{>0}\) and supported on \((1-\epsilon,1+\epsilon)\). Let \(\Phi_{1,\epsilon}(z)=e^{-{1\over 1-{z^{2}\over \epsilon^{2}}}}\), if \(z\in (-\epsilon,\epsilon)\) and \(\Phi_{1,\epsilon}(z)=0\) otherwise, \(z\in\mathcal{R}\), then \(\Phi_{1,\epsilon}\) is smooth on \(\mathcal{R}\) and supported on \((-\epsilon,\epsilon)\). Define \((\rho_{1},\overline{J})\) on \(\mathcal{R}^{2}\times\mathcal{R}\) by \(\rho_{1}(r,\theta,t)=\Phi_{\epsilon}(r)\rho(1,\theta,t)=\Phi_{\epsilon}(r)\Psi(\theta,t)\) for \(r>0, \theta\in [-\pi,\pi),t\in\mathcal{R}\), \(\rho_{1}(0,0,t)=0\), \(t\in\mathcal{R}\) ,\(\overline{J}(r,\theta,t)=\Phi_{\epsilon}(r)\overline{K}(1,\theta,t)=\Phi_{\epsilon}(r)J(\theta,t)(-\sin(\theta,\cos(\theta,0)\), for \(r>0, \theta\in [-\pi,\pi),t\in\mathcal{R}\), \(\overline{J}(0,0,t)=\overline{0}\), \(t\in\mathcal{R}\).

Then, using Lemma 2 and the facts that, for any given \(r>0\), \(\Phi_{\epsilon}(r)\Psi(\theta,t)\) and \(\Phi_{\epsilon}(r)J(\theta,t)\) satisfy the conditions of Lemma 1, we have that \((\rho_{1},\overline{J})\) satisfy the continuity equation on \(\mathcal{R}^{2}\times\mathcal{R}\).
Now define \((\rho_{1},\overline{J})\) on \(\mathcal{R}^{3}\times\mathcal{R}\) by \[\rho_{1}(r,\theta,z,t)=\Phi_{1,\epsilon}(z)\rho_{1}(r,\theta,t),\] for \(r>0, \theta\in [-\pi,\pi), z\in\mathcal{R},t\in\mathcal{R}\), \[\rho_{1}(0,0,z,t)=0,\;\; z\in \mathcal{R},\;\;t\in\mathcal{R},\] \[\overline{J}(r,\theta,z,t)=\Phi_{1,\epsilon}(z)\overline{J}(r,\theta,t),\] for \(r>0, \theta\in [-\pi,\pi), z\in\mathcal{R}, t\in\mathcal{R}\) \[\overline{J}(0,0,z,t)=\overline{0},\;\;z\in \mathcal{R},\;\;t\in\mathcal{R}.\] Clearly, \((\rho_{1},\overline{J})\) is supported on \(Ann(1,\epsilon)\times (-\epsilon,\epsilon)\), and if \(\overline{J}=(j_{1},j_{2},j_{3})\), we have that \(j_{3}(x,y,z,t)=0\), so that \({\partial j_{3}\over \partial z}=0\) and \[\label{lem6-eq2} \bigtriangledown\centerdot\overline{J}={\partial j_{1}\over \partial x}+{\partial j_{2}\over \partial y}. \tag{12}\] We have that, for any \(z\in\mathcal{R}\), \((\Phi_{1,\epsilon}(z)\rho_{1}(r,\theta,t),\Phi_{1,\epsilon}(z)\overline{J}(r,\theta,t))\) satisfies the continuity equation on \(\mathcal{R}^{2}\times\mathcal{R}\), so combining the result with (12), we obtain that \((\rho_{1},\overline{J})\) satisfies the continuity equation on \(\mathcal{R}^{3}\times\mathcal{R}\). ◻

Proof. By the proof in [8], we can find a pair \((\overline{E},\overline{B})\) in the base frame, with \(\square^{2}\overline{E}=\overline{0}\), and \(\overline{B}=\overline{0}\), (13), such that \((\rho,\overline{J},\overline{E},\overline{B})\) satisfy Maxwell’s equations. If \((\overline{E}',\overline{B}')\) is any pair such that \((\rho,\overline{J},\overline{E}',\overline{B}')\) satisfy Maxwell’s equations, then, taking the difference, \((0,\overline{0},\overline{E}-\overline{E}',\overline{B}-\overline{B}')\) is a vacuum solution to Maxwell’s equations, so that, see [2]; \[\square^{2}(\overline{E}-\overline{E}')=\overline{0},\qquad\square^{2}(\overline{B}-\overline{B}')=\overline{0}.\] From (13), we obtain that; \[\square^{2}\overline{E}'=\overline{0},\qquad \square^{2}\overline{B}'=\overline{0},\] as well. The last claim follows from the above proof and the results in [8]. ◻

Proof. We have that \(\overline{E}\times \overline{B}=(\overline{E}_{1}+\overline{E}_{2}+\overline{E}_{3})\times (\overline{B}_{1}+\overline{B}_{2})\). A simple calculation shows that, as \(c>1\), that for \(|\overline{r}|>1\), \(\overline{r}|\in S^{3}(r)\); \[\begin{aligned} |\overline{E}_{1}\times \overline{B}_{1}|_{\overline{r},t}&\leq {\mu_{0}\over 16\pi^{2}\epsilon_{0}} \int\limits_{S^{1}}\int\limits_{S^{1}} {\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|p|(\overline{r'},t-t_{r})|\overline{J}|(\overline{s'},t-t_{r}))\over {(r-1)}^{4}}d\theta d\phi\\ &\leq 4\pi^{2}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|p|(\overline{r'},t-t_{r})|\overline{J}|(\overline{s'},t-t_{r}))\over {(r-1)}^{4}},\\ |\overline{E}_{1}\times \overline{B}_{2}|&\leq {\mu_{0}\over 16\pi^{2}\epsilon_{0}} {\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|p|(\overline{r'},t-t_{r})|\dot{\overline{J}}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}}d\theta d\phi\\ &\leq 4\pi^{2}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|p|(\overline{r'},t-t_{r})|\dot{\overline{J}}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}},\\ |\overline{E}_{2}\times \overline{B}_{1}|&\leq {\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|\dot{p}|(\overline{r'},t-t_{r})\overline{J}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}}d\theta d\phi\\ &\leq 4\pi^{2}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|\dot{p}|(\overline{r'},t-t_{r})|\overline{J}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}},\\ |\overline{E}_{3}\times \overline{B}_{1}|&\leq {\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|\overline{J}|(\overline{r'},t-t_{r})|\dot{\overline{J}}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}}d\theta d\phi\\ &\leq 4\pi^{2}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}{\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|\overline{J}|(\overline{r'},t-t_{r})|\dot{\overline{J}}|(\overline{s'},t-t_{r}))\over {(r-1)}^{3}}. \end{aligned}\] It follows that, for \(r>1\); \[\begin{aligned} \max&\left(\left|\int\limits_{S^{3}(r)}(\overline{E}_{1}\times \overline{B}_{1}).d\overline{S}(r)|,\int\limits_{S^{3}(r)}(\overline{E}_{1}\times \overline{B}_{2}).d\overline{S}(r)|,\int\limits_{S^{3}(r)}(\overline{E}_{2}\times \overline{B}_{1}).d\overline{S}(r)|,\int\limits_{S^{3}(r)}(\overline{E}_{3}\times \overline{B}_{1}).d\overline{S}(r)\right|\right)\\ &=\max\left(\left|\int\limits_{S^{3}(r)}(\overline{E}_{1}\times \overline{B}_{1}).\hat{\overline{n}}dS(r)|,|\int\limits_{S^{3}(r)}(\overline{E}_{1}\times \overline{B}_{2}).\hat{\overline{n}}dS(r)|,|\int\limits_{S^{3}(r)}(\overline{E}_{2}\times \overline{B}_{1}).\hat{\overline{n}}dS(r)|,|\int\limits_{S^{3}(r)}(\overline{E}_{3}\times \overline{B}_{1}).\hat{\overline{n}}dS(r)\right|\right)\\ &\leq \max(\int\limits_{S^{3}(r)}|\overline{E}_{1}\times \overline{B}_{1})|dS(r)|,\int\limits_{S^{3}(r)}|(\overline{E}_{1}\times \overline{B}_{2})|dS(r),\int\limits_{S^{3}(r)}|(\overline{E}_{2}\times \overline{B}_{1})|dS(r),\int\limits_{S^{3}(r)}|(\overline{E}_{3}\times \overline{B}_{1})|dS(r)|)\\ &\leq \int\limits_{S^{3}(r)}(\max(|\overline{E}_{1}\times \overline{B}_{1}|(\overline{r},t),|\overline{E}_{1}\times \overline{B}_{2}|(\overline{r},t),|\overline{E}_{2}\times \overline{B}_{1}|(\overline{r},t),|\overline{E}_{3}\times \overline{B}_{1}|(\overline{r},t)))dS(r)\\ &\leq 4\pi^{2} {Area(\delta(S^{3}(r)))\over (r-1)^{3}}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}\max\limits_{\overline{r}',\overline{s}'\in S^{1}}\left(|p|\left(\overline{r}',t-t_{r}\right)|\overline{J}(\overline{s}',t-t_{r})|,|p|(\overline{r}',t-t_{r}) |\dot{\overline{J}}(\overline{s}',t-t_{r})|,|\dot{p}(\overline{r}',t-t_{r})|\right.\\ &\quad\times\left.|\overline{J}(\overline{s}',t-t_{r})|,|\overline{J}(\overline{r}', t-t_{r})||\dot{\overline{J}}((\overline{s}',t-t_{r}))|\right)\\ &=4\pi^{2}{4\pi r^{2}\over (r-1)^{3}}{\mu_{0}\over 16\pi^{2}\epsilon_{0}}\max\limits_{\overline{r}',\overline{s}'\in S^{1}}(|p|(\overline{r}',t-t_{r})|\overline{J}(\overline{s}',t-t_{r})|,|p|(\overline{r}',t-t_{r})|\dot{\overline{J}}(\overline{s}',t-t_{r})|,|\dot{p}(\overline{r}',t-t_{r})|\\ &\quad\times|\overline{J}(\overline{s}',t-t_{r})|,|\overline{J}(\overline{r}',t-t_{r})||\dot{\overline{J}}((\overline{s}',t-t_{r}))|)\\ &=C(r,t-t_{r}), \end{aligned}\] with \(\lim\limits_{r\rightarrow\infty}C(r,t-t_{r})=0\), for \(t\in\mathcal{R}\), given the assumption that \(lim_{t\rightarrow -\infty}\max\limits_{x\in S^{1}}(|p|,|\dot{p}|,|\overline{J}|,|\dot{\overline{J}})|_{x,t}\leq D\), with \(D\in\mathcal{R}\), see Remark 1.

It follows that, for \(t\in\mathcal{R}\), \[\lim\limits_{r\rightarrow\infty}P(r,t)=\lim\limits_{r\rightarrow\infty}\int\limits_{S^{3}(r)}(\overline{E}\times \overline{B})(\overline{r},t).d\overline{S}(r)=\lim\limits_{r\rightarrow\infty}\int\limits_{S^{3}(r)}(\overline{E}_{2}\times \overline{B}_{2}+\overline{E}_{3}\times \overline{B}_{2})(\overline{r},t).d\overline{S}(r),\] as required. ◻

Proof. Since, we have that \[\overline{E}_{2}(\overline{r},t)={1\over 4\pi\epsilon_{0}}\int\limits [{\dot{\rho}(\overline{r'},t_{r})\over c\mathfrak{r}}\hat{\mathfrak{\overline{r}}}]d\tau'\] where \(\overline{r}=(x,y,z)\), \(\overline{r'}=(\cos(\theta,\sin(\theta),0)\), \(\overline{r}-\overline{r'}=(x-\cos(\theta),y-\sin(\theta),z)\), \(\mathfrak{r}=|\overline{r}-\overline{r'}|=(x^{2}+y^{2}+z^{2}+1-2x\cos(\theta)-2y\sin(\theta))^{1\over 2}\) \(=(r^{2}+1-2x\cos(\theta)-2y\sin(\theta))^{1\over 2}\), \(\hat{\overline{\mathfrak{r}}}={(\overline{r}-\overline{r'})\over \mathfrak{r}}={(x-\cos(\theta),y-\sin(\theta),z)\over (x^{2}+y^{2}+z^{2}+1-2x\cos(\theta)-2y\sin(\theta))^{1\over 2}}\), \(\dot{\rho}(\theta,t)=-m\cos(m\theta)\sin(mt)\), \[\begin{aligned} \overline{E}_{2}\left(\overline{r},t\right)&={1\over 4\pi\epsilon_{0}c}\int\limits_{-\pi}^{\pi}{-m\cos\left(m\theta\right)\sin\left(mt_{r}\right)\over \left(r^{2}+1-2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\right)}\left(x-\cos\left(\theta\right),y-\sin\left(\theta\right),z\right)d\theta\\ &={1\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}{-m\cos\left(m\theta\right)\sin\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\over \left(1-{2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\over \left(r^{2}+1\right)}\right)}\left(x-\cos\left(\theta\right),y-\sin\left(\theta\right),z\right)d\theta\\ &={1\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}-m\cos\left(m\theta\right)\sin\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\left(x-\cos\left(\theta\right),y-\sin\left(\theta\right),z\right)d\theta+O\left({1\over r^{2}}\right)\\ &={1\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}-m\cos\left(m\theta\right)[\sin\left(mt\right)\cos\left(m{\mathfrak{r}\over c}\right)-\cos\left(mt\right)\sin\left(m{\mathfrak{r}\over c}\right)]\left(x,y,z\right)d\theta+O\left({1\over r^{2}}\right)\\ &={-m\sin\left(mt\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\cos\left(m{\mathfrak{r}\over c}\right)\left(x,y,z\right)d\theta +{m\cos\left(mt\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\sin\left(m{\mathfrak{r}\over c}\right)\left(x,y,z\right)d\theta\\ &\;\;\;+O\left({1\over r^{2}}\right) \end{aligned}\] Observe that, using Taylor expansions; \[\begin{aligned} \cos\left(m{\mathfrak{r}\over c}\right)&=\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\left(1+{x\cos\left(\theta\right)+y\sin\left(\theta\right)\over r^{2}+1}\right)\right)+O\left({1\over r}\right)\\ &=\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\\&\quad-\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)+O\left({1\over r}\right)\\ \sin\left(m{\mathfrak{r}\over c}\right)&=\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\left(1+{x\cos\left(\theta\right)+y\sin\left(\theta\right)\over r^{2}+1}\right)\right)+O\left({1\over r}\right)\\ &=\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\\ &\quad+\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)+O\left({1\over r}\right), \end{aligned}\] so that; \[ \label{lem9-eq1} \overline{E}_{2}\left(\overline{r},t\right)={-m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(x,y,z\right)d\theta\nonumber\\ +{m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(x,y,z\right)d\theta\nonumber\\ +{m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(x,y,z\right)d\theta\nonumber\\ +{m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(x,y,z\right)d\theta+O\left({1\over r^{2}}\right). \tag{15}\] We have that \[\begin{aligned} \cos\left(m\theta\right)&=Re\left(\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right)^{m}\right) =\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right), \end{aligned}\] \[\begin{aligned} \cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)&=\sum_{w=0}^{\infty}{\left(-1\right)^{w}m^{2w}\left(x\cos\left(\theta\right)+y\sin\left(\theta\right)\right)^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\\ &=\sum_{w=0}^{\infty}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}\cos^{2w-v}\left(\theta\right)y^{v}\sin^{v}\left(\theta\right), \end{aligned}\] \[\begin{aligned} \sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)&=\sum_{w=0}^{\infty}{\left(-1\right)^{w}m^{2w+1}\left(x\cos\left(\theta\right)+y\sin\left(\theta\right)\right)^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\\ &=\sum_{w=0}^{\infty}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}\cos^{2w+1-v}\left(\theta\right)y^{v}\sin^{v}\left(\theta\right), \end{aligned}\] so that; \[\begin{aligned} \overline{E}_{2}\left(\overline{r},t\right)=&{-m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\left(x,y,z\right)\\ &\times\int\limits_{-\pi}^{\pi}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta +{m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\\ &\times\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\left(x,y,z\right)\\ &\times\int\limits_{-\pi}^{\pi}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta +{m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\\ &\times\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\left(x,y,z\right) \end{aligned}\]\[\begin{aligned} &\times\int\limits_{-\pi}^{\pi}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta +{m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\\ &\times\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\left(x,y,z\right)\\ &\times\int\limits_{-\pi}^{\pi}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta+O\left({1\over r^{2}}\right). \end{aligned}\] We recall the result that; \[\begin{aligned} I_{\alpha}^{\beta}&=\int\limits_{-\pi}^{\pi}\cos^{\alpha}(\theta)\sin^{\beta}(\theta)d\theta={\pi\alpha!\beta!\over 2^{\alpha+\beta-1}({\alpha\over 2})!({\beta\over 2})!({\alpha+\beta\over 2})!},\quad\text{for $\alpha\geq 0$, $\beta\geq 0$, $\alpha$ and $\beta$ even,}\\ I_{\alpha}^{\beta}&=0,\quad\text{ otherwise}. \end{aligned}\] Applying the result in this case, we obtain; if \(m\) is even, then; \[ \label{lem9-eq2} \overline{E}_{2}\left(\overline{r},t\right){-m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\nonumber\\ \times\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\left(x,y,z\right)I_{2w+m-s-v}^{s+v}\nonumber\\ +{m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\nonumber\\ \times\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\left(x,y,z\right)I_{2w+m-s-v}^{s+v}+O\left({1\over r^{2}}\right), \tag{16}\] if \(m\) is odd, then; \[\ label{lem9-eq3} \overline{E}_{2}\left(\overline{r},t\right)={m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\nonumber\\ \times\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\left(x,y,z\right)I_{2w+1+m-s-v}^{s+v}\nonumber\\ +{m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\nonumber\\ \times\sum_{v=0,v, even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\left(x,y,z\right)I_{2w+1+m-s-v}^{s+v}+O\left({1\over r^{2}}\right). \tag{17}\] We have that \[\begin{aligned} \overline{E}_{3}\left(\overline{r},t\right)&={-1\over 4\pi\epsilon_{0}}\int\limits \left[{\dot{\overline{J}}\left(\overline{r'},t_{r}\right)\over c^{2}\mathfrak{r}}\right]d\tau'\dot{\overline{J}}\left(\theta,t\right)\\ &=m\sin\left(m\theta\right)\cos\left(mt\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right), \end{aligned}\] \[\ label{lem9-eq4} \overline{E}_{3}\left(\overline{r},t\right)={-1\over 4\pi\epsilon_{0}c^{2}}\int\limits_{-\pi}^{\pi}{m\sin\left(m\theta\right)\cos\left(mt_{r}\right)\over \left(r^{2}+1-2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\right)^{1\over 2}}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ {-1\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}{m\sin\left(m\theta\right)\cos\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\over \left(1-{2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\over \left(r^{2}+1\right)}\right)^{1\over 2}}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ {-1\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}m\sin\left(m\theta\right)\cos\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta+O\left({1\over r^{2}}\right)\nonumber\\ {-1\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}m\sin\left(m\theta\right)[\cos\left(mt\right)\cos\left(m{\mathfrak{r}\over c}\right)+\sin\left(mt\right)\sin\left(m{\mathfrak{r}\over c}\right)]\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ +O\left({1\over r^{2}}\right)\nonumber\\ {-m\cos\left(mt\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\cos\left(m{\mathfrak{r}\over c}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ -{m\sin\left(mt\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left(m{\mathfrak{r}\over c}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta+O\left({1\over r^{2}}\right)\nonumber\\ {-m\cos\left(mt\right)\cos\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ +{m\cos\left(mt\right)\sin\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ -{m\sin\left(mt\right)\sin\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ -{m\sin\left(mt\right)\cos\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)d\theta\nonumber\\ +O\left({1\over r^{2}}\right). \tag{18}\] We have that \[\sin\left(m\theta\right)=Im\left(\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right)^{m}\right)=\sum_{s\leq m,s odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right),\] so that \[\begin{aligned} \overline{E}_{3}\left(\overline{r},t\right)=&{-m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\\ &\times\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\int\limits_{-\pi}^{\pi}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\\ &+{m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}} \end{aligned}\]\[\begin{aligned} &\times\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\int\limits_{-\pi}^{\pi}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\\ &-{m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\\ &\times\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\int\limits_{-\pi}^{\pi}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\\ &-{m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\\ &\times\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\int\limits_{-\pi}^{\pi}\left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta+O\left({1\over r^{2}}\right). \end{aligned}\] It follows that; for \(m\) even \[\ label{lem9-eq5} \overline{E}_{3}\left(\overline{r},t\right)+{m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\nonumber\\ \times\left(-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+2+m-v-s}^{v+s},0\right)\nonumber\\ -{m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\nonumber\\ \times\left(-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+2+m-v-s}^{v+s},0\right)\nonumber\\ +O\left({1\over r^{2}}\right), \tag{19}\] and for \(m\) odd; \[\ overline{E}_{3}\left(\overline{r},t\right) {-m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\nonumber\\ \times\left(-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+1+m-v-s}^{v+s},0\right)\nonumber\\ -{m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi\epsilon_{0}c^{2}\left(r^{2}+1\right)^{1\over 2}}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\nonumber\\ \times\left(-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+1+m-v-s}^{v+s},0\right)\nonumber\\ +O\left({1\over r^{2}}\right). \tag{20}\] We have that; \[\begin{aligned} \overline{B}_{2}\left(\overline{r},t\right)&={\mu_{0}\over 4\pi}\int\limits \left[{\dot{\overline{J}}\left(\overline{r'},t_{r}\right)\over c\mathfrak{r}}\right]\times \hat{\mathfrak{\overline{r}}} d\tau' \left(-\sin\left(\theta\right),\cos\left(\theta\right),0\right)\times \left(x-\cos\left(\theta\right),y-\sin\left(\theta\right),z\right)\\ &=\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)\left(y-\sin\left(\theta\right)-\cos\left(\theta\right)\left(x-\cos\left(\theta\right)\right)\right), \end{aligned}\] \[\begin{aligned} %\label{lem9-eq7} \overline{B}_{2}\left(\overline{r},t\right)&={\mu_{0}\over 4\pi c}\int\limits_{-\pi}^{\pi}{m\sin\left(m\theta\right)\cos\left(mt_{r}\right)\over \left(r^{2}+1-2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\right)}\notag\\ &\quad\times\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)\left(y -\sin\left(\theta\right)-\cos\left(\theta\right)\left(x-\cos\left(\theta\right)\right)\right)\right)d\theta\notag\\ &={\mu_{0}\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}{m\sin\left(m\theta\right)\cos\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\over \left(1-{2x\cos\left(\theta\right)-2y\sin\left(\theta\right)\over \left(r^{2}+1\right)}\right)}\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta+O\left({1\over r^{2}}\right)\notag\\ &={\mu_{0}\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi} m\sin\left(m\theta\right)\cos\left(m\left(t-{\mathfrak{r}\over c}\right)\right)\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta+O\left({1\over r^{2}}\right)\notag\\ &={\mu_{0}m\cos\left(mt\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi} \sin\left(m\theta\right)\cos\left(m{\mathfrak{r}\over c}\right)\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta\notag\\ &\quad+{\mu_{0}m\sin\left(mt\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi} \sin\left(m\theta\right)\sin\left(m{\mathfrak{r}\over c}\right)\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta+O\left({1\over r^{2}}\right)\notag\\ &={\mu_{0}m\cos\left(mt\right)\cos\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\notag\\ &\quad\times\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta\notag\\ &\quad-{\mu_{0}m\cos\left(mt\right)\sin\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right) \notag\\ &\quad\times\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta\notag\\ &\quad+{\mu_{0}m\sin\left(mt\right)\sin\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\cos\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\notag\\ &\quad\times\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta\notag\\ &\quad+{\mu_{0}m\sin\left(mt\right)\cos\left(m\left(r^{2}+1\right)^{1\over 2}\right)\over 4\pi c\left(r^{2}+1\right)}\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\notag\\ &\quad\times\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta+O\left({1\over r^{2}}\right)\notag\\ &={\mu_{0}m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\notag\\ &\quad\times\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\int\limits_{-\pi}^{\pi}\left(z\cos\left(\theta\right),z\sin\left(\theta\right),-y\sin\left(\theta\right)-x\cos\left(\theta\right)\right)\notag\\ &\quad\times\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\notag\\ &\quad-{\mu_{0}m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\notag \end{aligned}\] \[\ quad\times\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\int\limits_{-\pi}^{\pi}\left(z\cos\left(\theta\right),z\sin\left(\theta\right),-y\sin\left(\theta\right)-x\cos\left(\theta\right)\right)\notag\\ \quad\times\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\notag\\ \quad+{\mu_{0}m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\notag\\ \quad\times\sum_{v=0}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\int\limits_{-\pi}^{\pi}\left(z\cos\left(\theta\right),z\sin\left(\theta\right),-y\sin\left(\theta\right)-x\cos\left(\theta\right)\right)\notag\\ \quad\times\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta\notag\\ \quad+{\mu_{0}m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\notag\\ \quad\times\sum_{v=0}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\int\limits_{-\pi}^{\pi}\left(z\cos\left(\theta\right),z\sin\left(\theta\right),-y\sin\left(\theta\right)-x\cos\left(\theta\right)\right)\notag\\ \quad\times\cos^{m-s}\left(\theta\right)\sin^{s}\left(\theta\right)\cos^{2w+1-v}\left(\theta\right)\sin^{v}\left(\theta\right)d\theta+O\left({1\over r^{2}}\right). \tag{21}\] It follows that; for \(m\) even; \[ \overline{B}_{2}\left(\overline{r},t\right)-{\mu_{0}m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\notag\\ \times\left(\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+2+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+1+m-v-s}^{s+v+1}\right.\notag\\ \left.-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v+1}I_{2w+1+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+2-v}y^{v}I_{2w+2+m-v-s}^{s+v}\right)\notag\\ +{\mu_{0}\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\notag\\ \times\left(\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+2+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+1+m-v-s}^{s+v+1}\right.\notag\\ \left.-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v+1}I_{2w+1+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+2-v}y^{v}I_{2w+2+m-v-s}^{s+v}\right)+O\left({1\over r^{2}}\right), \tag{22}\] and for \(m\) odd; \[\begin{aligned} %\label{lem9-eq9} \overline{B}_{2}\left(\overline{r},t\right)=&{\mu_{0}m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\notag\\ &\times\left(\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+1+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+m-v-s}^{s+v+1}\right.\notag\\ &\left.-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v+1}I_{2w+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{s+v}\right)\notag \end{aligned}\] \[\ +{\mu_{0}\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\over 4\pi c\left(r^{2}+1\right)}\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\notag\\ \times\left(\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+1+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+m-v-s}^{s+v+1}\right.\notag\\ \left.-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v+1}I_{2w+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{s+v}\right)+O\left({1\over r^{2}}\right). \tag{23}\] We now compute the Poynting vectors \(\overline{E}_{2}\times \overline{B}_{2}\) and \(\overline{E}_{3}\times \overline{B}_{2}\) in the cases when \(m\) is even and \(m\) is odd. If \(m\) is even, by (15) and (16) we have that; \[\overline{E}_{2,e}=\overline{E}_{2,e}^{1}+\overline{E}_{2,e}^{2}=-\alpha m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma+\alpha m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma,\] where \(\alpha={1\over 4\pi\epsilon_{0}c(r^{2}+1)}\) and \(\Gamma=\int\limits_{-\pi}^{\pi}\cos\left(m\theta\right)\cos\left({mx\cos(\theta)+my\sin(\theta)\over \left(r^{2}+1\right)^{1\over 2}}\right)(x,y,z)d\theta\). If \(m\) is even, by (21) and (22); \[\overline{B}_{2,e}=\overline{B}_{2,e}^{1}+\overline{B}_{2,e}^{2} =-\beta m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'+\beta m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'+O\left({1\over r^{2}}\right),\] where \(\beta={\mu_{0}\over 4\pi c(r^{2}+1)}\) and \(\Gamma'=\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(\cos\left(\theta\right)z,\sin\left(\theta\right)z,-\sin\left(\theta\right)y-\cos\left(\theta\right)x\right)d\theta\). It follows that; \[\begin{aligned} \overline{E}_{2,e}\times \overline{B}_{2,e} =&\overline{E}_{2,e}^{1}\times \overline{B}_{2,e}^{1}+\overline{E}_{2,e}^{2}\times \overline{B}_{2,e}^{1}+\overline{E}_{2,e}^{1}\times\overline{B}_{2,e}^{2}+\overline{E}_{2,e}^{2}\times\overline{B}_{2,e}^{2}+O\left({1\over r^{3}}\right)\\ =&\alpha\beta m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma\times\Gamma'\\ &-\alpha\beta m^{2}\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma\times\Gamma'-\alpha\beta m^{2}\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma\times\Gamma'\\ &+\alpha\beta m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma\times\Gamma'+O\left({1\over r^{3}}\right)\\ =&\alpha\beta m^{2}[-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\\ &+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)]\Gamma\times\Gamma'+O\left({1\over r^{3}}\right). \end{aligned}\] Similarly, by (18) and (19); \[\overline{E}_{3,e}=\overline{E}_{3,e}^{1}+\overline{E}_{3,e}^{2} =\gamma m\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''-\gamma m\sin\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''+O\left({1\over r^{2}}\right),\] where \(\gamma={1\over 4\pi\epsilon_{0}c^{2}(r^{2}+1)^{1\over 2}}\) and \(\Gamma''=\int\limits_{-\pi}^{\pi}\sin\left(m\theta\right)\sin\left({mx\cos\left(\theta\right)+my\sin\left(\theta\right)\over \left(r^{2}+1\right)^{1\over 2}}\right)\left(-\sin(\theta),\cos(\theta),0\right)d\theta\). If \(m\) is even, it follows that; \[\begin{aligned} \overline{E}_{3,e}\times \overline{B}_{2,e} =&\overline{E}_{3,e}^{1}\times \overline{B}_{2,e}^{1}+\overline{E}_{3,e}^{2}\times \overline{B}_{2,e}^{1}+\overline{E}_{3,e}^{1}\times\overline{B}_{2,e}^{2}+\overline{E}_{3,e}^{2}\times\overline{B}_{2,e}^{2}+O\left({1\over r^{3}}\right) \end{aligned}\]\[\begin{aligned} =&-\beta\gamma m^{2}\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''\times\Gamma'\\ &+\beta\gamma m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''\times\Gamma'\\ &+\beta\gamma m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''\times\Gamma'\\ &-\beta\gamma m^{2}\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''\times\Gamma'+O\left({1\over r^{3}}\right)\\ =&\beta\gamma m^{2}[-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\\ &+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)]\Gamma''\times\Gamma'+O\left({1\over r^{3}}\right), \end{aligned}\] If \(m\) is odd, by (15) and (17) we have that; \[\overline{E}_{2,o}=\overline{E}_{2,o}^{1}+\overline{E}_{2,o}^{2}+O\left({1\over r^{2}}\right) =\alpha m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''+\alpha m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''',\] where \(\Gamma'''=\int\limits_{-\pi}^{\pi}\cos(m\theta)\sin\left({mx\cos(\theta)+my\sin(\theta)\over (r^{2}+1)^{1\over 2}}\right)(x,y,z)d\theta\). If \(m\) is odd, by (21) and (23); \[\overline{B}_{2,o}=\overline{B}_{2,o}^{1}+\overline{B}_{2,o}^{2}+O\left({1\over r^{2}}\right) =\beta m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''''+\beta m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma''''+O\left({1\over r^{2}}\right),\] where \(\Gamma''''=\int\limits_{-\pi}^{\pi}\sin(m\theta)\cos\left({mx\cos(\theta)+my\sin(\theta)\over (r^{2}+1)^{1\over 2}}\right)(\cos(\theta)z,\sin(\theta)z,-\sin(\theta)y-\cos(\theta)x)d\theta\). It follows that; \[\begin{aligned} \overline{E}_{2,o}\times \overline{B}_{2,o} =&\overline{E}_{2,o}^{1}\times \overline{B}_{2,o}^{1}+\overline{E}_{2,o}^{2}\times \overline{B}_{2,o}^{1}+\overline{E}_{2,o}^{1}\times\overline{B}_{2,o}^{2}+\overline{E}_{2,o}^{2}\times\overline{B}_{2,o}^{2}+O\left({1\over r^{3}}\right)\\ =&\alpha\beta m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''\times\Gamma''''\\ &+\alpha\beta m^{2}\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''\times\Gamma''''\\ &+\alpha\beta m^{2}\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''\times\Gamma''''\\ &+\alpha\beta m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''\times\Gamma''''+O\left({1\over r^{3}}\right)\\ &=\alpha\beta m^{2}[\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)+\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\\ &+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)]\Gamma'''\times\Gamma''''+O\left({1\over r^{3}}\right). \end{aligned}\] Similarly, by (18) and (20); \[\overline{E}_{3,o}=\overline{E}_{3,o}^{1}+\overline{E}_{3,o}^{2} =-\gamma m\cos\left(mt\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''-\gamma m\sin\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''+O\left({1\over r^{2}}\right),\] where \(\Gamma'''''=\int\limits_{-\pi}^{\pi}\sin(m\theta)\cos\left({mx\cos(\theta)+my\sin(\theta)\over (r^{2}+1)^{1\over 2}}\right)(-\sin(\theta),\cos(\theta),0)d\theta\). If \(m\) is odd, it follows that; \[\begin{aligned} \overline{E}_{3,o}\times \overline{B}_{2,o} =&\overline{E}_{3,o}^{1}\times \overline{B}_{2,o}^{1}+\overline{E}_{3,o}^{2}\times \overline{B}_{2,o}^{1}+\overline{E}_{3,o}^{1}\times\overline{B}_{2,o}^{2}+\overline{E}_{3,o}^{2}\times\overline{B}_{2,o}^{2}+O\left({1\over r^{3}}\right)\\ =&-\beta\gamma m^{2}\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''\times\Gamma''''\\ &-\beta\gamma m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''\times\Gamma''''\\ &-\beta\gamma m^{2}\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''\times\Gamma''''\\ &-\beta\gamma m^{2}\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\Gamma'''''\times\Gamma''''+O\left({1\over r^{3}}\right)\\ =&\beta\gamma m^{2}[-\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\\ &-2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)]\Gamma'''''\times\Gamma''''+O\left({1\over r^{3}}\right). \end{aligned}\] We compute \(\Gamma\times\Gamma'\). By (16) and (22), we have that; \[\Gamma=\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}} \sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}\left(x,y,z\right)I_{2w+m-s-v,s+v},\] \[\begin{aligned} \Gamma'\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\\ &\times\left(\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+2+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}zI_{2w+1+m-v-s}^{s+v+1}\right.\\ &\left.-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v+1}I_{2w+1+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+2-v}y^{v}I_{2w+2+m-v-s}^{s+v}\right), \end{aligned}\] so that; \[\begin{aligned} %\label{lem9-eq101} \Gamma\times\Gamma'=&\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s, even,s'\leq m,s',odd}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\left(-1\right)^{s'-1\over 2}C^{m}_{s'}{\left(-1\right)^{w'}m^{2w'+1}\over \left(2w'+1\right)!\left(r^{2}+1\right)^{w'+{1\over 2}}}\nonumber\\ &\left(-\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'+2}I_{2w+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'+1}\right.\nonumber\\ &-\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+m-s-v}^{s+v}I_{2w'+2-m-v'-s'}^{s'+v'}\nonumber\\ &-\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}z^{2}I_{2w+m-s-v}^{s+v}I_{2w'+1-m-v'-s'}^{s'+v'+1},\nonumber \end{aligned}\] \[\ +\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'+1}\nonumber\\ +\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+3-v-v'}y^{v+v'}I_{2w+m-s-v}^{s+v}I_{2w'+2+m-v'-s'}^{s'+v'}\nonumber\\ +\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}z^{2}I_{2w+m-s-v}^{s+v}I_{2w'+2+m-v'-s'}^{s'+v'},\nonumber\\ +\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'}zI_{2w+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'+1}\nonumber\\ \left.-\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'+1}C^{2w}_{v}C^{2w'+1}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'+1}zI_{2w+m-s-v}^{s+v}I_{2w'+2+m-v'-s'}^{s'+v'}\right). \tag{24}\] By (19), we have that; \[\begin{aligned} \Gamma''=&\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\\ &\times\left(-\sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}I_{2w+2+m-v-s}^{v+s},0\right). \end{aligned}\] It follows that; \[\ Gamma''\times\Gamma'\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\left(-1\right)^{s'-1\over 2}C^{m}_{s'}{\left(-1\right)^{w'}m^{2w'+1}\over \left(2w'+1\right)!\left(r^{2}+1\right)^{w'+{1\over 2}}}\nonumber\\ \left(-\sum_{v=0,v,odd,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+2+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'+1}\right.\nonumber\\ -\sum_{v=0,v,odd,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+3-v-v'}y^{v+v'}I_{2w+2+m-s-v}^{s+v}I_{2w'+2-m-v'-s'}^{s'+v'},\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+1+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'+1}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+3-v-v'}y^{v+v'}I_{2w+1+m-s-v}^{s+v+1}I_{2w'+2+m-v'-s'}^{s'+v'},\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'}zI_{2w+1+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'+1}\nonumber\\ \left.-\sum_{v=0,v,odd,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'}zI_{2w+1++m-s-v}^{s+v+1}I_{2w'+2+m-v'-s'}^{s'+v'}\right). \tag{25}\] We now compute the flux of the Poynting vectors over the sphere \(S(r)\). We have that \(x=r\sin(\phi)\cos(\theta)\), \(y=r\sin(\phi)\sin(\theta)\), \(z=r\cos(\phi)\) with coordinates \(0\leq \phi<\pi\) and \(-\pi\leq \theta<\pi\). We have that; \[\begin{aligned} r_{\phi}\times r_{\theta}&=(r\cos(\phi)\cos(\theta),r\cos(\phi)\sin(\theta),-r\sin(\phi))\times (-r\sin(\phi)\sin(\theta),r\sin(\phi)\cos(\theta),0)\\ &=r^{2}(\sin^{2}(\phi)\cos(\theta),\sin^{2}(\phi)\sin(\theta),\sin(\phi)\cos(\phi), \end{aligned}\] so that; \[d\overline{S}=\hat{\overline{n}}dS=r^{2}(\sin^{2}(\phi)\cos(\theta),\sin^{2}(\phi)\sin(\theta),\sin(\phi)\cos(\phi))d\phi d\theta,\] and \[\int\limits_{S(r)}(\Gamma\times\Gamma')\centerdot d\overline{S} =\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}(\Gamma\times\Gamma')|_{(r\sin(\phi)\cos(\theta),r\sin(\phi)\sin(\theta),r\cos(\phi))}\centerdot r^{2}(\sin^{2}(\phi)\cos(\theta), \sin^{2}(\phi)\sin(\theta),\sin(\phi)\cos(\phi)d\theta d\phi.\] Applying this to (24) gives; \[\begin{align} \int\limits_{S\left(r\right)}\left(\Gamma\times\Gamma’\right)\centerdot d\overline{S}&=\sum_{w=0}^{\infty}\sum_{w’=0}^{\infty}\sum_{s\leq m,s, even,s’\leq m,s’,odd}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\left(-1\right)^{s’-1\over 2}C^{m}_{s’}{\left(-1\right)^{w’}m^{2w’+1}\over \left(2w’+1\right)!\left(r^{2}+1\right)^{w’+{1\over 2}}}\nonumber\\ &\quad\times\left(-\sum_{v=0,v,even,v’=0,v’,even}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’+1}\right.\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’+2}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,odd}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+2-m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,even}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+1-m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad+\sum_{v=0,v,even,v’=0,v’,even}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’+2}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad+\sum_{v=0,v,even,v’=0,v’,odd}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+2+m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad+\sum_{v=0,v,even,v’=0,v’,odd}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+2+m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+1-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad+\sum_{v=0,v,even,v’=0,v’,even}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,odd}^{2w,2w’+1}C^{2w}_{v}C^{2w’+1}_{v’}I_{2w+m-s-v}^{s+v}I_{2w’+2+m-v’-s’}^{s’+v’}\nonumber\\ &\quad\left.\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+1-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\right). \end{align}\tag{26}\]

An inspection of (26) shows that the terms 1 and 4, 2 and 5, 3 and 7, and 6 and 8 cancel. This proves that; \[\int\limits_{S(r)}(\Gamma\times\Gamma')\centerdot d\overline{S}=0.\] Applying the same method to (25), we have that; \[\begin{align} \int\limits_{S\left(r\right)}\left(\Gamma”\times\Gamma’\right)\centerdot d\overline{S}&=\sum_{w=0}^{\infty}\sum_{w’=0}^{\infty}\sum_{s\leq m,s,odd,s’\leq m,s’,odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\left(-1\right)^{s’-1\over 2}C^{m}_{s’}{\left(-1\right)^{w’}m^{2w’+1}\over \left(2w’+1\right)!\left(r^{2}+1\right)^{w’+{1\over 2}}}\nonumber\\ &\quad-\sum_{v=0,v,odd,v’=0,v’,even}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+2+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,odd,v’=0,v’,odd}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+2+m-s-v}^{s+v}I_{2w’+2-m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+4-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+1+m-s-v}^{s+v+1}I_{2w’+1+m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’+2}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,odd}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+1+m-s-v}^{s+v+1}I_{2w’+2+m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+1+m-s-v}^{s+v+1}I_{2w’+1+m-v’-s’}^{s’+v’+1}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &\quad-\sum_{v=0,v,odd,v’=0,v’,odd}^{2w+1,2w’+1}C^{2w+1}_{v}C^{2w’+1}_{v’}I_{2w+1++m-s-v}^{s+v+1}I_{2w’+2+m-v’-s’}^{s’+v’}\nonumber\\ &\quad\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi. \end{align}\tag{27}\]

Using the notation \(I_{\alpha}^{\beta}\) again, and letting \(J_{\gamma}={2^{\gamma+1}({\gamma-1\over 2})!^{2}\over \gamma!}=\int\limits_{0}^{\pi}\sin^{\gamma}(\phi)d\phi\) for \(\gamma\) odd, we obtain that; \[\begin{aligned} \int\limits_{S(r)}&(\Gamma''\times\Gamma')\centerdot d\overline{S} =\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w+1}\over (2w+1)!(r^{2}+1)^{w+{1\over 2}}}\\ &\times(-1)^{s'-1\over 2}C^{m}_{s'}{(-1)^{w'}m^{2w'+1}\over (2w'+1)!(r^{2}+1)^{w'+{1\over 2}}}r^{2w+2w'+5}\\ &\times\left(-\sum_{v=0,v,odd,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+2+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'+1}I_{2w+2w'+3-v-v'}^{v+v'+1}J_{2w+2w'+5}\right.\\ &-\sum_{v=0,v,odd,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+2+m-s-v}^{s+v}I_{2w'+2-m-v'-s'}^{s'+v'}I_{2w+2w'+4-v-v'}^{v+v'}J_{2w+2w'+5}\\ &-\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+1+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'+1}I_{2w+2w'+2-v-v'}^{v+v'+2}J_{2w+2w'+5} \end{aligned}\]\[\begin{aligned} &-\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+1+m-s-v}^{s+v+1}I_{2w'+2+m-v'-s'}^{s'+v'}I_{2w+2w'+3-v-v'}^{v+v'+1}J_{2w+2w'+5}\\ &-\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+1+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'+1}I_{2w+2w'+2-v-v'}^{v+v'}(J_{2w+2w'+3}-J_{2w+2w'+5})\\ &\left.-\sum_{v=0,v,odd,v'=0,v',odd}^{2w+1,2w'+1}C^{2w+1}_{v}C^{2w'+1}_{v'}I_{2w+1++m-s-v}^{s+v+1}I_{2w'+2+m-v'-s'}^{s'+v'}I_{2w+2w'+2-v-v'}^{v+v'}(J_{2w+2w'+3}-J_{2w+2w'+5})\right)\\ =&\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w+1}\over (2w+1)!(r^{2}+1)^{w+{1\over 2}}}\\ &\times(-1)^{s'-1\over 2}C^{m}_{s'}{(-1)^{w'}m^{2w'+1}\over (2w'+1)!(r^{2}+1)^{w'+{1\over 2}}}r^{2w+2w'+5}c_{w,w',s,s',m}, \end{aligned}\] where we have abbreviated the term in brackets to \(c_{w,w',s,s',m}<0\). We compute \(\Gamma'''\times \Gamma''''\). By (17) and (23); \[\ Gamma'''=\sum_{w=0}^{\infty}\sum_{s\leq m,s, even}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}} \sum_{v=0,v,even}^{2w+1}C^{2w+1}_{v}x^{2w+1-v}y^{v}\left(x,y,z\right)I_{2w+1+m-s-v}^{s+v}\] and \[\begin{aligned} \Gamma''''=&\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\\ &\times\left(\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+1+m-v-s}^{s+v},\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}zI_{2w+m-v-s}^{s+v+1}\right.\\ &\left.-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v+1}I_{2w+m-v-s}^{s+v+1}-\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w+1-v}y^{v}I_{2w+1+m-v-s}^{s+v}\right), \end{aligned}\] so that; \[\ Gamma'''\times\Gamma'''' \sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s, even,s'\leq m,s',odd}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\left(-1\right)^{s'-1\over 2}C^{m}_{s'}{\left(-1\right)^{w'}m^{2w'}\over \left(2w'\right)!\left(r^{2}+1\right)^{w'}}\nonumber\\ \times\left(-\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'+2}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}\right.\nonumber\\ -\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}z^{2}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1},\nonumber\\ +\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'+1}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ +\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+3-v-v'}y^{v+v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}\nonumber\\ +\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}z^{2}I_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'},\nonumber\\ +\sum_{v=0,v,even,v'=0,v',even}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+2-v-v'}y^{v+v'}zI_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ \left.-\sum_{v=0,v,even,v'=0,v',odd}^{2w+1,2w'}C^{2w+1}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'+1}zI_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}\right). \tag{28}\]

Integrating (28), we obtain; \[\begin{align}\label{lem9-eq106} \int\limits_{S\left(r\right)}\left(\Gamma”’\times\Gamma””\right)\centerdot d\overline{S}=&\sum_{w=0}^{\infty}\sum_{w’=0}^{\infty}\sum_{s\leq m,s, even,s’\leq m,s’,odd}\left(-1\right)^{s\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\left(-1\right)^{s’-1\over 2}C^{m}_{s’}{\left(-1\right)^{w’}m^{2w’}\over \left(2w’\right)!\left(r^{2}+1\right)^{w’}}\nonumber\\ &\times\left(-\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+m-v’-s’}^{s’+v’+1}\right.\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’+2}\left(\theta\right)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,even,v’=0,v’,odd}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+m-v’-s’}^{s’+v’+1}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &+\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+m-v’-s’}^{s’+v’+1}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’+2}\left(\theta\right)d\theta d\phi\nonumber\\ &+\sum_{v=0,v,even,v’=0,v’,odd}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+5}\left(\phi\right)\cos^{2w+2w’+3-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &+\sum_{v=0,v,even,v’=0,v’,odd}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+1-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\nonumber\\ &+\sum_{v=0,v,even,v’=0,v’,even}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+m-v’-s’}^{s’+v’+1}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+2-v-v’}\left(\theta\right)\sin^{v+v’}\left(\theta\right)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,even,v’=0,v’,odd}^{2w+1,2w’}C^{2w+1}_{v}C^{2w’}_{v’}I_{2w+1+m-s-v}^{s+v}I_{2w’+1+m-v’-s’}^{s’+v’}\nonumber\\ &\left.\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w’+5}\sin^{2w+2w’+3}\left(\phi\right)\cos^{2}\left(\phi\right)\cos^{2w+2w’+1-v-v’}\left(\theta\right)\sin^{v+v’+1}\left(\theta\right)d\theta d\phi\right). \end{align}\tag{29}\]

An inspection of (29) shows that the terms 1 and 4, 2 and 5, 3 and 7, and 6 and 8 cancel again. This proves that; \[\int\limits_{S(r)}(\Gamma'''\times\Gamma'''')\centerdot d\overline{S}=0.\] By (20), we have that; \[\begin{aligned} \Gamma'''''=&\sum_{w=0}^{\infty}\sum_{s\leq m,s, odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w}\over (2w)!(r^{2}+1)^{w}}\\ &\times\left(-\sum_{v=0,v,even}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+m-v-s}^{v+s+1},\sum_{v=0,v,odd}^{2w}C^{2w}_{v}x^{2w-v}y^{v}I_{2w+1+m-v-s}^{v+s},0\right). \end{aligned}\] It follows that; \[\ Gamma'''''\times\Gamma''''=\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w}\over (2w)!(r^{2}+1)^{w}}(-1)^{s'-1\over 2}C^{m}_{s'}{(-1)^{w'}m^{2w'}\over (2w')!(r^{2}+1)^{w'}}\nonumber\\ \times\left(-\sum_{v=0,v,odd,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'-v-v'}y^{v+v'+1}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}\right.\nonumber\\ -\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1-m-v'-s'}^{s'+v'},\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'-v-v'}y^{v+v'+1}I_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'+1-v-v'}y^{v+v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'},\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'-v-v'}y^{v+v'}zI_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ \left.-\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}x^{2w+2w'-v-v'}y^{v+v'}zI_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}\right). \tag{30}\] Integrating (30), we get that; \[\begin{aligned} %\label{lem9-eq108} \int\limits_{S(r)}(\Gamma'''''\times\Gamma'''')\centerdot d\overline{S}=&\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w}\over (2w)!(r^{2}+1)^{w}}(-1)^{s'-1\over 2}C^{m}_{s'}{(-1)^{w'}m^{2w'}\over (2w')!(r^{2}+1)^{w'}}\nonumber\\ &\times\left(-\sum_{v=0,v,odd,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}\right.\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+3}(\phi)\cos^{2w+2w'+1-v-v'}(\theta)\sin^{v+v'+1}(\theta)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1-m-v'-s'}^{s'+v'}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+3}(\phi)\cos^{2w+2w'+2-v-v'}(\theta)\sin^{v+v'}(\theta)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+3}(\phi)\cos^{2w+2w'-v-v'}(\theta)\sin^{v+v'+2}(\theta)d\theta d\phi\nonumber\\ &-\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'}\nonumber\\ &\times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+3}(\phi)\cos^{2w+2w'+1-v-v'}(\theta)\sin^{v+v'+1}(\theta)d\theta d\phi\nonumber \end{aligned}\] \[\ label{lem9-eq108} -\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}\nonumber\\ \times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+1}(\phi)\cos^{2}(\phi)\cos^{2w+2w'-v-v'}(\theta)\sin^{v+v'}(\theta)d\theta d\phi\nonumber\\ \left.-\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}\right)\nonumber\\ \times\int\limits_{0}^{\pi}\int\limits_{-\pi}^{\pi}r^{2w+2w'+3}\sin^{2w+2w'+1}(\phi)\cos^{2}(\phi)\cos^{2w+2w'-v-v'}(\theta)\sin^{v+v'}(\theta)d\theta d\phi\notag\\ \sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w}\over (2w)!(r^{2}+1)^{w}}(-1)^{s'-1\over 2}C^{m}_{s'}{(-1)^{w'}m^{2w'}\over (2w')!(r^{2}+1)^{w'}}r^{2w+2w'+3}\nonumber\\ \times\left(-\sum_{v=0,v,odd,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+m-v'-s'}^{s'+v'+1}I_{2w+2w'+1-v-v'}^{v+v'+1}J_{2w+2w'+3}\right.\nonumber\\ -\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1-m-v'-s'}^{s'+v'}I_{2w+2w'+2-v-v'}^{v+v'}J_{2w+2w'+3}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}I_{2w+2w'-v-v'}^{v+v'+2}J_{2w+2w'+3}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+1+m-v'-s'}^{s'+v'}I_{2w+2w'+1-v-v'}^{v+v'+1}J_{2w+2w'+3}\nonumber\\ -\sum_{v=0,v,even,v'=0,v',even}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+m-s-v}^{s+v+1}I_{2w'+m-v'-s'}^{s'+v'+1}I_{2w+2w'-v-v'}^{v+v'}(J_{2w+2w'+1}-J_{2w+2w'+3})\nonumber\\ \left.-\sum_{v=0,v,odd,v'=0,v',odd}^{2w,2w'}C^{2w}_{v}C^{2w'}_{v'}I_{2w+1+m-s-v}^{s+v}I_{2w'+1+m-v'-s'}^{s'+v'}I_{2w+2w'-v-v'}^{v+v'}(J_{2w+2w'+1}-J_{2w+2w'+3})\right)\nonumber\\ \sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}(-1)^{s-1\over 2}C^{m}_{s}{(-1)^{w}m^{2w}\over (2w)!(r^{2}+1)^{w}}(-1)^{s'-1\over 2}\nonumber\\ \times C^{m}_{s'}{(-1)^{w'}m^{2w'}\over (2w')!(r^{2}+1)^{w'}}r^{2w+2w'+3}d_{w,w',s,s',m}, \tag{31}\] where again we have denoted the term in brackets by \(d_{w,w',s,s',m}\). We conclude that, if \(m\) is even; \[\begin{aligned} P\left(r,t\right)=&\int\limits_{S\left(r\right)}\left(E_{2,e}\times B_{2,e}\right)\centerdot d\overline{S}+\int\limits_{S\left(r\right)}\left(E_{3,e}\times B_{2,e}\right)\centerdot d\overline{S}\\ =&\int\limits_{S\left(r\right)}\left(\left[\alpha\beta m^{2}\left[-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\right.\right.\\ &\left.\left.\left.+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right]\Gamma\times\Gamma'\right]+O\left({1\over r^{3}}\right)\right)\centerdot d\overline{S}\\ &+\int\limits_{S\left(r\right)}\left(\beta\gamma m^{2}\left[-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\right.\\ &\left.\left.+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right]\Gamma''\times\Gamma'+O\left({1\over r^{3}}\right)\right)\centerdot d\overline{S}\\ =&\beta\gamma m^{2}\left[-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right. \end{aligned}\] \[\begin{aligned} &\left.+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right]\\ &\times\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w+1}\over \left(2w+1\right)!\left(r^{2}+1\right)^{w+{1\over 2}}}\\ &\times\left(-1\right)^{s'-1\over 2}C^{m}_{s'}{\left(-1\right)^{w'}m^{2w'+1}\over \left(2w'+1\right)!\left(r^{2}+1\right)^{w'+{1\over 2}}}r^{2w+2w'+5}c_{w,w',s,s',m}+O\left({1\over r}\right). \end{aligned}\] Similarly, if \(m\) is odd, we obtain that; \[\begin{aligned} P\left(r,t\right)=&\int\limits_{S\left(r\right)}\left(E_{2,o}\times B_{2,o}\right)\centerdot d\overline{S}+\int\limits_{S\left(r\right)}\left(E_{3,o}\times B_{2,o}\right)\centerdot d\overline{S}\\ =&\int\limits_{S\left(r\right)}\left(E_{3,o}\times B_{2,o}\right)\centerdot d\overline{S}\\ =&\beta\gamma m^{2}\left[-\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &\left.-2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right]\\ &\times\sum_{w=0}^{\infty}\sum_{w'=0}^{\infty}\sum_{s\leq m,s,odd,s'\leq m,s',odd}\left(-1\right)^{s-1\over 2}C^{m}_{s}{\left(-1\right)^{w}m^{2w}\over \left(2w\right)!\left(r^{2}+1\right)^{w}}\left(-1\right)^{s'-1\over 2}\\ &\times C^{m}_{s'}{\left(-1\right)^{w'}m^{2w'}\over \left(2w'\right)!\left(r^{2}+1\right)^{w'}}r^{2w+2w'+3}d_{w,w',s,s',m}+O\left({1\over r}\right). \end{aligned}\] ◻

Proof. It is easy to see, replacing summations over even indices, with odd indices, and vice versa, following the proof of the previous lemma, and assuming \(m\) is even, that \(E^{1}_{2}=c(r,m,t)\Gamma\), \(E^{2}_{2}=d(r,m,t)\Gamma\), \(E^{3}_{2}=e(r,m,t)\Delta\), \(E^{4}_{2}=f(r,m,t)\Delta\) where \(\Gamma=\int\limits_{-\pi}^{\pi}\cos(m\theta)\cos\left({mx\cos(\theta)+my\sin(\theta)\over (r^{2}+1)^{1\over 2}}\right)(x,y,z)d\theta\) and \(\Delta=\int\limits_{-\pi}^{\pi}\sin(m\theta)\cos\left({mx\cos(\theta)+my\sin(\theta)\over (r^{2}+1)^{1\over 2}}\right)(x,y,z)d\theta\) and \(\{c,d,e,f\}\) are parameters not depending on \(\theta,x,y\) or \(z\). A similar argument works for \(\Gamma'\) and \(\Delta'\) as in the statement of the lemma. It is therefore sufficient to check that \(\int\limits_{S(r)}(\Gamma\times\Gamma')\centerdot d\overline{S}=\int\limits_{S(r)}(\Gamma\times\Delta')\centerdot d\overline{S}=\int\limits_{S(r)}(\Delta\times\Gamma')\centerdot d\overline{S}=\int\limits_{S(r)}(\Delta\times\Delta')\centerdot d\overline{S}=0\)

The first case was checked in the previous lemma. The remaining cases follow from the first case, by replacing even with odd summations in both \(v\) and \(s\) when replacing \(\Gamma\) by \(\Delta\), and, similarly, for the pair \(\Gamma'\) and \(\Delta'\) in \(v'\) and \(s'\). For the remainder of the lemma, carefully follow the calculation in the previous result, the details are left to the reader. ◻

Proof. The proof is a simple calculation, using the result of the previous lemma. ◻

Proof. We consider linear combinations \(a_{1}\rho_{1}+a_{2}\rho_{2}+a_{3}\rho_{3}+a_{4}\rho_{4}\) and \(a_{1}J_{1}+a_{2}J_{2}+a_{3}J_{3}+a_{4}J_{4}\) , where \(\{a_{1},a_{2},a_{3},a_{4}\}\) are real scalars. Let \(\{E_{m,comb},B_{m,comb}\}\) denote the resulting fields obtained from Jefimenko’s equations. We have, by linearity, that \(E_{2,m,comb}=\sum_{i=1}^{4}E_{2}^{i}\), \(E_{3,m,comb}=\sum_{i=1}^{4}E_{3}^{i}\),\(B_{2,m,comb}=\sum_{i=1}^{4}B_{2}^{i}\) and computing the power radiated through a sphere of radius \(r\), using lemmas 10 and 11; \[\begin{aligned} P\left(r,t\right)&=\int\limits_{S\left(r\right)}\left(E_{m,comb}\times B_{m,comb}\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=\int\limits_{S\left(r\right)}\left(\left(E_{2,m,comb}+E_{3,m,comb}\right)\times B_{2,m,comb}\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=\int\limits_{S\left(r\right)}\left(E_{2,m,comb}\times B_{2,m,comb}\right)\centerdot d\overline{S}+\int\limits_{S\left(r\right)}\left(E_{3,m,comb}\times B_{2,m,comb}\right)\centerdot d\overline{S}+O\left({1\over r}\right) \end{aligned}\]\[\begin{aligned} &=\sum_{i,j=1}^{4}a_{i}a_{j}\int\limits_{S\left(r\right)}\left(E_{2}^{i}\times B_{2}^{j}\right)\centerdot d\overline{S}+\sum_{i=1}^{4}a_{i}a_{j}\int\limits_{S\left(r\right)}\left(E_{3}^{i}\times B_{2}^{j}\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=\sum_{i,j=1}^{4}a_{i}a_{j}\int\limits_{S\left(r\right)}\left(E_{3}^{i}\times B_{2}^{j}\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=\left(a_{1}^{2}C_{1}+a_{2}^{2}C_{2}-2a_{1}a_{2}C_{3}\right)\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'\right)\centerdot d\overline{S}+\left(a_{3}^{2}C_{1}+a_{4}^{2}C_{2}-2a_{3}a_{4}C_{3}\right)\int\limits_{S\left(r\right)}\left(\Delta''\times\Delta'\right)\centerdot d\overline{S}\\ &+\left(-a_{1}a_{3}C_{1}-a_{2}a_{4}C_{2}+\left(a_{2}a_{3}+a_{1}a_{4}\right)C_{3}\right)\int\limits_{S\left(r\right)}\left(\Delta''\times\Gamma'\right)\centerdot d\overline{S}\\ &+\left(-a_{1}a_{3}C_{1}-a_{2}a_{4}C_{2}+\left(a_{1}a_{4}+a_{2}a_{3}\right)C_{3}\right)\int\limits_{S\left(r\right)}\left(\Gamma''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right) \end{aligned}\] We note the identities \[\begin{aligned} C_{1}+C_{2}&=\beta\gamma m^{2}\left(-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &\left.-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)+\beta\gamma m^{2}\left(-\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &\left.-2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\\ &=\beta\gamma m^{2}\left(-\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\\ &=-\beta\gamma m^{2} \end{aligned}\] and; \[\begin{aligned} C_{1}-C_{2}&=\beta\gamma m^{2}\left(-\cos^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)+2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &\left.-\sin^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\beta\gamma m^{2}\left(-\sin^{2}\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &-\left.2\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos^{2}\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\\ &=-\beta\gamma m^{2}\left(\cos\left(2mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ &\left.+4\sin\left(mt\right)\cos\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\cos\left(2mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\\ &=\beta\gamma m^{2}\left(\cos\left(2mt\right)\left(\cos\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\sin\left(2mt\right)\sin\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\\ \int\limits_{t}^{t+{\pi\over m}}C_{3}dt&=\int\limits_{t}^{t+{\pi\over m}}\left(\beta\gamma m^{2}\left(\cos\left(mt\right)\sin\left(mt\right)\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\right.\\ &\left.\left.-\sin\left(mt\right)\cos\left(mt\right)\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-\sin^{2}\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\right. \end{aligned}\] \[\begin{aligned} &\left.\left.+\cos^{2}\left(mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\right)dt\\ &=\int\limits_{t}^{t+{\pi\over m}}\left(\beta\gamma m^{2}\left({\sin\left(2mt\right)\over 2}\sin^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)-{\sin\left(2mt\right)\over 2}\cos^{2}\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right.\right.\\ &\left.\left.+\cos\left(2mt\right)\sin\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\cos\left(m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)\right)dt\\ &=0 \end{aligned}\] so we can obtain a convenient simplification by requiring that;

1. \(a_{1}^{2}=a_{2}^{2}\),

2. \(a_{3}^{2}=a_{4}^{2}\),

3. \(a_{1}a_{2}=-a_{3}a_{4}\),

4. \(a_{1}a_{3}=-a_{2}a_{4}\),

5. \(a_{2}a_{3}=-a_{1}a_{4}\).

The last \(2\) conditions give that \(a_{1}={-a_{2}a_{4}\over a_{3}}\) (\(a_{3}\neq 0\)), \(a_{2}a_{3}=-{-a_{2}a_{4}\over a_{3}}a_{4}\), \(a_{3}^{2}=a_{4}^{2}\) (\(a_{3}\neq 0\)), and \(a_{3}={-a_{1}a_{4}\over a_{2}}\) (\(a_{2}\neq 0\)), \(a_{1}-{-a_{1}a_{4}\over a_{2}}=-a_{2}a_{4}\), \(a_{1}^{2}=a_{2}^{2}\) (\(a_{4}\neq 0\)), which are conditions (i) and (ii). Conversely, if \(a_{1}=a_{2}\) and \(a_{3}=-a_{4}\) or \(a_{1}=-a_{2}\) and \(a_{3}=a_{4}\), then conditions (i),(ii),(iv),(v) are satisfied. Substituting into condition (iii), we then require that; \(a_{1}^{2}=a_{3}^{2}\), which we can achieve if; \(a_{1}=a_{3}\) or \(a_{1}=-a_{3}\). We then obtain that; \[\begin{aligned} P\left(r,t\right)=&-a_{1}^{2}\beta\gamma m^{2}\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'\right)\centerdot d\overline{S}-a_{3}^{2}\beta\gamma m^{2}\int\limits_{S\left(r\right)}\left(\Delta''\times\Delta'\right)\centerdot d\overline{S} -2a_{1}a_{2}C_{3}\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'-\Delta''\times\Delta'\right)\centerdot d\overline{S}\\ &-a_{1}a_{3}\beta\gamma m^{2} \cos\left(2mt\right)\left(\cos\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\sin\left(2mt\right)\sin\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\int\limits_{S\left(r\right)}\left(\Delta''\times\Gamma'\right)\centerdot d\overline{S}\\ &-a_{1}a_{3}\beta\gamma m^{2}\cos\left(2mt\right)\left(\cos\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\sin\left(2mt\right)\sin\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\int\limits_{S\left(r\right)}\left(\Gamma''\times\Delta'\right)\centerdot d\overline{S}\\ &+O\left({1\over r}\right). \end{aligned}\] If we integrate through a period of \(\pi\over m\), we obtain that; \[\begin{aligned} \int\limits_{t}^{t+{\pi\over m}}P\left(r,t\right)dt &=-a_{1}^{2}\beta\gamma m^{2}{\pi\over m}\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=-\pi a_{1}^{2}\beta\gamma m\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right), \end{aligned}\] as \[\int\limits_{t}^{t+{\pi\over m}}\cos\left(2mt\right)dt=\int\limits_{t}^{t+{\pi\over m}}\sin\left(2mt\right)dt=\int\limits_{t}^{t+{\pi\over m}}C_{3}dt=0.\] By Poynting’s Theorem, we have that; \[{dW\over dt}=-{d\over dt}\int\limits_{B\left(r\right)}{1\over 2}\left(\epsilon_{0}E^{2}+{1\over \mu_{0}}B^{2}\right)dB\left(r\right)-{1\over \mu_{0}}P\left(r,t\right).\] Integrating this expression over a period of \({\pi\over m}\) and using periodicity, from Jefimenko’s equations that \(\overline{E}\left(\overline{r},t+{\pi\over m}\right)=-\overline{E}\left(\overline{r},t\right)\), and \(\overline{B}\left(\overline{r},t+{\pi\over m}\right)=-\overline{B}\left(\overline{r},t\right)\), we obtain that; \[\begin{aligned} W\left(t+{\pi\over m}\right)-W\left(t\right)&=-\int\limits_{B\left(r\right)}{1\over 2}\left(\epsilon_{0}E^{2}+{1\over \mu_{0}}B^{2}\right)dB\left(r\right)|_{t+{\pi\over m}}+\int\limits_{B\left(r\right)}{1\over 2}\left(\epsilon_{0}E^{2}+{1\over \mu_{0}}B^{2}\right)dB\left(r\right)|_{t}-{1\over \mu_{0}}\int\limits_{t}^{t+{\pi\over m}}P\left(r,t\right)\\ &={1\over \mu_{0}}\pi a_{1}^{2}\beta\gamma m\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right). \end{aligned}\] It follows that, for \(r>0\); \[\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}=\int\limits_{B\left(r\right)}\left(\bigtriangledown\centerdot\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\right)dB\left(r\right)={16\pi c^{3}c\left(t,m\right)\epsilon_{0}\left(r^{2}+1\right)^{3\over 2}\over a_{1}^{2}m}+O\left(r^{2}\right),\] where \(c(t,m)=W(t+{\pi\over m})-W(t)\), the difference in mechanical energy of the charge/current distribution confined to a vanishing annulus containing \(S^{1}\), is independent of \(r>1\), as contained in \(B(r)\), (³). Letting \(f(x,y,z)=(\bigtriangledown\centerdot(\Gamma''\times\Gamma'+\Delta''\times\Delta'))\), we obtain that; \[\int\limits_{B(r)}f dB(r)=d(t,m)\left(r^{2}+1\right)^{3\over 2}+O(r^{2}),\] where \(d(t,m)\) is independent of \(r\). Dividing by \(r^{3}\), we obtain that; \[\lim_{r\rightarrow \infty}{1\over r^{3}}\int\limits_{B(r)}f dB(r)=d(t,m)+O\left({1\over r}\right).\] Taking a power series expansion of \(f\) in the variables \(\{x,y,z\}\), and integrating term by term, we can see that \(f\) must be constant. Using the volume of the ball \(B(r)\) as \({4\pi r^{3}\over 3}\), we must have that \(f=d(t,m)=c(t,m)=0\). It follows, letting \(r\rightarrow\infty\), that the mechanical energy of the matter wave doesn’t vary over a cycle and moreover that the power radiated \(\int\limits_{t}^{t+m}P(r,t)dt\) over a ball \(B(r)\) and a cycle is \(O({1\over r})\), for any \(r>0\), and \(\lim\limits_{r\rightarrow\infty}\int\limits_{t}^{t+{\pi\over m}}P(r,t)dt=0\). ◻

Proof. The first result is clear. If we denote a solution \(\Psi_{new}\) to (32) by \(\Psi(x,ct)\), and let \(J_{new}=cJ(x,ct)\), with corresponding \(\{\rho_{new},\overline{J}_{new}\}\), then in the Jefimenko equations, we get that; \[\begin{aligned} \overline{E}_{2,new}(x,t)&=c\overline{E}_{2}\left[\rho_{new},{\overline{J}_{new}\over c}\right],\\ \overline{E}_{3,new}&=c^{2}\overline{E}_{2}\left[\rho_{new},{\overline{J}_{new}\over c}\right],\\ \overline{B}_{2,new}&=c^{2}\overline{B}_{2}\left[\rho_{new},{\overline{J}_{new}\over c}\right]. \end{aligned}\] Again, we obtain that \[\int\limits_{S(r)}\left(\overline{E}_{2,new}\times \overline{B}_{2,new}\right)\centerdot d\overline{S}=c^{3}\int\limits_{S(r)}\left(\overline{E}_{2}\times \overline{B}_{2}\right)\left[\rho_{new},{\overline{J}_{new}\over c}\right]\centerdot d\overline{S}=0,\] and; \[\int\limits_{S(r)}\left(\overline{E}_{3,new}\times \overline{B}_{2,new}\right)\centerdot d\overline{S}=c^{4}\int\limits_{S(r)}\left(\overline{E}_{3}\times \overline{B}_{2}\right)\left[\rho_{new},{\overline{J}_{new}\over c}\right]\centerdot d\overline{S},\] so that, following Lemma 8; \[P^{new}(r,t)+O\left({1\over r}\right)=c^{4}P(r,t)\left[\rho_{new},{\overline{J}_{new}\over c}\right]+O\left({1\over r}\right).\] We then obtain, for the linear combination in Lemma 12 that; \[\begin{aligned} P^{new}\left(r,t\right)=&c^{4}\left[-a_{1}^{2}\beta\gamma m^{2}\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'\right)\centerdot d\overline{S}-a_{3}^{2}\beta\gamma m^{2}\int\limits_{S\left(r\right)}\left(\Delta''\times\Delta'\right)\centerdot d\overline{S} -2a_{1}a_{2}C_{3}'\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'-\Delta''\times\Delta'\right)\centerdot d\overline{S}\right.\\ &-a_{1}a_{3}\beta\gamma m^{2}\cos\left(2mct\right)\left(\cos\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\sin\left(2mct\right)\sin\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\int\limits_{S\left(r\right)}\left(\Delta''\times\Gamma'\right)\centerdot d\overline{S}\\ &\left.-a_{1}a_{3}\beta\gamma m^{2}\cos\left(2mct\right)\left(\cos\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\right)-\sin\left(2mct\right)\sin\left(2m{\left(r^{2}+1\right)^{1\over 2}\over c}\right)\int\limits_{S\left(r\right)}\left(\Gamma''\times\Delta'\right)\centerdot d\overline{S}\right]\\ &+O\left({1\over r}\right), \end{aligned}\] where \(C_{3}'=C_{3}(ct)\). If we integrate through a period of \(\pi\over mc\), we obtain that; \[\begin{aligned} \int\limits_{t}^{t+{\pi\over mc}}P^{new}\left(r,t\right)dt&=-a_{1}^{2}\beta\gamma m^{2}{c^{4}\pi\over mc}\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right)\\ &=-c^{3}\pi a_{1}^{2}\beta\gamma m\int\limits_{S\left(r\right)}\left(\Gamma''\times\Gamma'+\Delta''\times\Delta'\right)\centerdot d\overline{S}+O\left({1\over r}\right). \end{aligned}\] Now we can repeat the argument to obtain the same result, that \(\lim\limits_{r\rightarrow\infty}\int\limits_{t}^{t+{\pi\over mc}}P^{new}(r,t)dt=0\). ◻

Proof. We clearly have that \({\partial \rho_{0}\over \partial t}+{\partial J_{0}\over \partial x}=0\), so that when we extend \(J_{0}\) to \(S^{1}\) by \(\overline{J}_{0}(1,\theta)=J_{0}(\theta)(-\sin(\theta),\cos(\theta),0)\), for \(-\pi\leq \theta<\pi\), we have for the further extension \((\rho_{0},\overline{J}_{0})\) on \(Ann(1,\epsilon)\times (-\epsilon,\epsilon)\), using Lemma 6, that \({\partial \rho_{0}\over \partial t}+\bigtriangledown\centerdot\overline{J}_{0}=0\). Using Lemma 8, we have that; \[\lim\limits_{r\rightarrow\infty}P(r)=\lim\limits_{r\rightarrow\infty}\int\limits_{S(r)}(\overline{E}_{0,2}\times\overline{B}_{0,2}+\overline{E}_{0,3}\times\overline{B}_{0,2})d\overline{S}(r),\] but as \(\dot{\rho_{0}}=0\) and \(\dot{\overline{J}_{0}}=\overline{0}\), we have that \(\overline{E}_{0,2}=\overline{E}_{0,3}=\overline{B}_{0,2}=\overline{0}\), so that \(\lim\limits_{r\rightarrow\infty}P(r)=0\) and \(\{\overline{E}_{0},\overline{B}_{0}\}\) satisfy the no radiating condition. Again, we have that \({\partial (\rho+\rho_{0})\over \partial t}+{\partial (J+J_{0})\over \partial x}=0\), with the same remark on the extension, so that \({\partial (\rho+\rho_{0})\over \partial t}+\bigtriangledown\centerdot(\overline{J}+\overline{J}_{0})=0\). We have that \(\overline{E}_{1}=\overline{E}+\overline{E}_{0}\) and \(\overline{B}_{1}=\overline{B}+\overline{B}_{0}\), so that; \[\begin{aligned} \lim_{r\rightarrow\infty}P\left(r\right)&=\lim_{r\rightarrow\infty}\int\limits_{S\left(r\right)}\left(\overline{E}_{1}\times\overline{B}_{1}\right)d\overline{S}\left(r\right)\\ &=\lim_{r\rightarrow\infty}\int\limits_{S\left(r\right)}\left(\left(\overline{E}+\overline{E}_{0}\right)\times\left(\overline{B}+\overline{B}_{0}\right)\right)d\overline{S}\left(r\right)\\ &=\lim_{r\rightarrow\infty}\left(\int\limits_{S\left(r\right)}\left(\overline{E}\times \overline{B}\right)d\overline{S}\left(r\right)+\int\limits_{S\left(r\right)}\left(\overline{E}_{0}\times \overline{B}_{0}\right)d\overline{S}\left(r\right) +\int\limits_{S\left(r\right)}\left(\overline{E}\times \overline{B}_{0}\right)d\overline{S}\left(r\right)+\int\limits_{S\left(r\right)}\left(\overline{E}_{0}\times \overline{B}\right)d\overline{S}\left(r\right)\right)\\ &=\lim_{r\rightarrow\infty}\left(\int\limits_{S\left(r\right)}\left(\overline{E}\times \overline{B}_{0}\right)d\overline{S}\left(r\right)+\int\limits_{S\left(r\right)}\left(\overline{E}_{0}\times \overline{B}\right)d\overline{S}\left(r\right)\right)\\ &=\lim_{r\rightarrow\infty}\left(\int\limits_{S\left(r\right)}\left(\overline{E}_{2}\times \overline{B}_{0,2}\right)d\overline{S}\left(r\right)+\int\limits_{S\left(r\right)}\left(\overline{E}_{3}\times \overline{B}_{0,2}\right)d\overline{S}\left(r\right)\right)\\ &\quad +\lim_{r\rightarrow\infty}\left(\int\limits_{S\left(r\right)}\left(\overline{E}_{0,2}\times \overline{B}_{2}\right)d\overline{S}\left(r\right)+\int\limits_{S\left(r\right)}\left(\overline{E}_{0,3}\times \overline{B}_{2}\right)d\overline{S}\left(r\right)\right)\\ &=0, \end{aligned}\] as required. ◻

Proof. We have that; \[\begin{aligned} T(\overline{x},t)|_{S^{1}}&={|\overline{J}(\overline{x},t)|\over |\rho(\overline{x},t)|}\\ &=a\left|{a_{1}\sin(mx)\sin(mt)-a_{2}\sin(mx)\cos(mt)-a_{2}\cos(mx)\sin(mt)-a_{1}\cos(mx)\cos(mt)\over a_{1}\cos(mx)\cos(mt)+a_{2}\cos(mx)\sin(mt)+a_{2}\sin(mx)\cos(mt)-a_{1}\sin(mx)\sin(mt)}\right|\\ &=|-1|\\ &=1, \end{aligned}\] where \(a=|(-\sin(\theta),\cos(\theta),0)|=1\), when \(\rho(\overline{x},t)\neq 0\), so that, using Lemma 6, for any configuration \((\rho,\overline{J})\), satisfying the continuity equation and extending \((\rho,J)\), we have \(T(\overline{x},t)|_{S^{1}}=1\). ◻

Proof. Suppose that there exists a frame \(S_{\overline{v}}\) with the property that the transformed fields \((\overline{E}_{\overline{v}},\overline{B}_{\overline{v}})\) is classically radiating in a cycle. We can therefore assume that there exists a sequence of times \(\{t_{i}:i\in\mathcal{Z}\}\), such that for sufficiently large \(r\in\mathcal{R}_{>0}\), we have that; \[\int_{B(\overline{0},r)}div(\overline{E}_{\overline{v}}\times \overline{B}_{\overline{v}})dB=f_{i,i+1}(t),\] for some continuous function \(f_{i,i+1}\) on \([t_{i},t_{i+1}]\), \(i\in\mathcal{Z}\), with \(f_{i,i+1}(t_{i})=f_{i,i+1}(t_{i+1})=0\) and \(|\int_{t_{i}}^{t_{i+1}}f_{i,i+1}dt|>\epsilon\), \(i\in\mathcal{Z}\). By Poynting’s Theorem, we have that; \[{d\over dt}\int_{B(\overline{0},r)}(u_{mech}+u_{em})dB=-\int_{B(\overline{0},r)}div(\overline{E}_{\overline{v}}\times \overline{B}_{\overline{v}})dB=-f_{i,i+1}(t),\] in the period \((t_{i},t_{i+1})\), so that, by the fundamental theorem of calculus; \[\int_{B(\overline{0},r)}(u_{mech}+u_{em})dB|_{t_{i+1}}-\int_{B(\overline{0},r)}(u_{mech}+u_{em})dB|_{t_{i}}=-\int_{t_{i}}^{t_{i+1}}f_{i,i+1}(t)dt.\] It follows that, if \(\int_{t_{i}}^{t_{i+1}}f_{i,i+1}(t)dt>0\), \(\int_{B(\overline{0},r)}(u_{mech}+u_{em})dB\) would decrease. Without loss of generality we can assume this as we can consider a time reversal; \[(-\rho_{\overline{v}}(\overline{x},t_{0}-t),\overline{J}_{\overline{v}}(\overline{x},t_{0}-t),-\overline{E}_{\overline{v}}(\overline{x},t_{0}-t), \overline{B}_{\overline{v}}(\overline{x},t_{0}-t)),\] which satisfies Maxwell’s equations with the flux reversed. Then as \(\sum_{i\in\mathcal{Z}}\int_{t_{i}}^{t_{i+1}}f_{i,i+1}(t)dt=\infty\) and \(\int_{B(\overline{0},r)}(u_{mech}+u_{em})dB\) is finite, we can assume that that there exists \(s_{r}\in\mathcal{R}_{>0}\), with \(u_{mech}|_{B(\overline{0},r)}=0\), either for \(t\geq s_{r}\), or for \(t\leq -s_{r}\), \((*)\). As \(u_{mech}\) is finite, and the property \((*)\) can be made uniform in \(r\), for \(r\) sufficiently large, we can assume no charge flows to infinity. Then, we can assume, if \(u_{mech}=0\) in \(B(\overline{0},r)\), that \(\rho|_{B(\overline{0},r)}\neq 0\), and as local kinetic energy of the charge is zero, that the local speed \(|\overline{w}|\) is zero, so that \({|\overline{J}_{\overline{v}}|\over |\rho_{\overline{v}}|}=|\overline{w}|\rightarrow 0\). In particularly the components \({j_{i,\overline{v}}\over \rho_{\overline{v}}}\rightarrow 0\). By the transformation laws back to the base frame, assuming, without loss of generality, that \(\overline{v}=(v,0,0)\), we have that; \[{|\overline{J}|\over |\rho|}={|\gamma_{v}(\overline{J}_{\overline{v},||}+\gamma_{v} \overline{v}\rho_{\overline{v}})+\overline{J}_{\overline{v},\perp}|\over |\gamma_{v}(\rho_{\overline{v}}+{\gamma_{v} vJ_{\overline{v},||}\over c^{2}})|} ={|(\gamma_{v} j_{1,v}+\gamma_{v} v\rho_{v},j_{2,v},j_{3,v})|\over |\gamma_{v} \rho_{v}+{\gamma_{v} vj_{1,v}\over c^{2}}|} ={|(\gamma_{v} {j_{1,v}\over \rho_{v}} +\gamma_{v}v,{j_{2,v}\over \rho_{v}},{j_{3,v}\over \rho_{v}})|\over |\gamma_{v}+{\gamma_{v}vj_{1,v}\over \rho_{v}c^{2}}|} \rightarrow {|\gamma_{v}v|\over |\gamma_{v}|}=v.\] However, this contradicts Lemma 15 unless \(v=1\). We can then obtain the result by a limiting argument to exclude this special case. ◻