Finite multipole expansion with an absolute dipole-power error

Dipole moment identities. The compact-ball verification in the preceding transverse-radiation theorem [F1] applies to the same $a>0$ and Countable Choice hypothesis: ${\bar{B}}_a$ is closed and bounded, hence compact; it is Borel and Lebesgue measurable; and ${\lambda }_3({\bar{B}}_a)\le {\lambda }_3(B_{2a})<\mathrm{\infty }$. Thus it is an admissible fixed integration set for [F3]. For each $i\in \{1,2,3\}$ put $p_i(u)={\int }_{B_a}{y}_i\rho (y,u)dy$. Smoothness and vanishing of $\rho ,J$ for $\mathrm{\mid }y\mathrm{\mid }>a$ imply that $\rho$, $J_i$ and their time derivatives vanish on $\mathrm{\partial }{B}_a$. Thus the integrals of $y_i\rho$ and $y_i{\rho }_t$ over $B_a$ agree with those over that compact measurable finite-measure set ${\bar{B}}_a$. Apply the set form of [F3] with integration set ${\bar{B}}_a$ and parameter $u$ to $f_i(y,u)=y_i\rho (y,u)$; its parameter derivative is $y_i{\rho }_t(y,u)$, and both are continuous on a neighborhood of ${\bar{B}}_a$ times any compact time interval. This gives $p_i^{\mathrm{\prime }}(u)={\int }_{B_a}{y}_i{\rho }_t(y,u)dy$. Fix $R_0>a$ and apply the divergence theorem to $y\mapsto y_{i}J(y,u)$ on $B_{R_0}$. The boundary flux is zero because $J=0$ near $\mathrm{\partial }{B}_{R_0}$, while $\mathrm{div}(y_{i}J)=J_i+y_i\mathrm{div}J$. Using ${\rho }_t=-\mathrm{div}J$ and the vanishing of the source outside $B_a$ gives $p_i^{\mathrm{\prime }}(u)={\int }_{B_a}{J}_i(y,u)dy$. For each $i$, the integrals of $J_i$ and $J_{i,t}$ over $B_a$ agree with those over ${\bar{B}}_a$; the closed ball has the compactness, measurability and finite measure just verified using [F1]. Both integrands are continuous on a neighborhood of ${\bar{B}}_a$ times any compact time interval. Apply the set form of [F3] on ${\bar{B}}_a$ separately to each component $J_i$ with parameter $u$; differentiating the preceding identity gives $p_i^{\mathrm{\prime }\mathrm{\prime }}(u)={\int }_{B_a}{J}_{i,t}(y,u)dy$. Collecting the three component identities yields $p^{\mathrm{\prime }\mathrm{\prime }}(u)={\int }_{B_a}{J}_t(y,u)dy$.

Finiteness of the remainder. For fixed $u$, define $g_u(y):={\max }_{\mathrm{\mid }s-u\mathrm{\mid }\le a\mathrm{/}c}\mathrm{\mid }{\mathrm{\partial }}_t^{N+2}J(y,s)\mathrm{\mid }$. The maximum exists and is finite by [F8]. For $y,y^{\mathrm{\prime }}\in {\bar{B}}_a$, ${\displaystyle \mathrm{\mid }{g}_u(y)-g_u(y^{\mathrm{\prime }})\mathrm{\mid }\le \underset{\mathrm{\mid }s-u\mathrm{\mid }\le a\mathrm{/}c}{\sup }\mid \mathrm{\mid }{\mathrm{\partial }}_t^{N+2}J(y,s)\mathrm{\mid }-\mathrm{\mid }{\mathrm{\partial }}_t^{N+2}J(y^{\mathrm{\prime }},s)\mathrm{\mid }\mid \le \underset{\mathrm{\mid }s-u\mathrm{\mid }\le a\mathrm{/}c}{\sup }\mathrm{\mid }{\mathrm{\partial }}_t^{N+2}J(y,s)-{\mathrm{\partial }}_t^{N+2}J(y^{\mathrm{\prime }},s)\mathrm{\mid }\mathrm{.}}$ Uniform continuity in [F8] makes the last expression tend to zero as $y^{\mathrm{\prime }}\to y$. Thus $g_u$ is continuous and bounded on ${\bar{B}}_a$, hence integrable on $B_a$; this proves that ${\mathcal{R}}_N(u)$ is finite.

Finite Taylor expansion. Fix $n\in S^2$, $u\in K$, and $y\in B_a$, and put $\tau =n\cdot y\mathrm{/}c$, so $\mathrm{\mid }\tau \mathrm{\mid }\le a\mathrm{/}c$. Apply [F4] to the curve $s\mapsto J_t(y,s)$ with Taylor order $N$ at $u$ and increment $\tau$. It gives ${\displaystyle J_t(y,u+\tau )=\sum _{k=0}^N\frac{{\tau }^k}{k!}{\mathrm{\partial }}_t^{k+1}J(y,u)+R_{N+1}(y),\mathrm{\mid }{R}_{N+1}(y)\mathrm{\mid }\le \frac{\mathrm{\mid }\tau {\mathrm{\mid }}^{N+1}}{(N+1)!}\underset{\mathrm{\mid }s-u\mathrm{\mid }\le a\mathrm{/}c}{\sup }\mathrm{\mid }{\mathrm{\partial }}_t^{N+2}J(y,s)\mathrm{\mid }\mathrm{.}}$ Integrate this identity over $B_a$. Linearity identifies the left integral with $V(n,u)$ and the polynomial integral with ${\mathcal{J}}_N(n,u)$. The integral triangle inequality and $\mathrm{\mid }\tau \mathrm{\mid }\le a\mathrm{/}c$ then give $\mathrm{\mid }V(n,u)-{\mathcal{J}}_N(n,u)\mathrm{\mid }\le {\mathcal{R}}_N(u)$. Since ${\mathcal{J}}_N$ is a finite polynomial in $n$ with finite source moments, it is continuous on the compact sphere and $A_N(u)<\mathrm{\infty }$.

Field remainder. By the transverse estimate of [F1], ${\displaystyle \mid E(rn,u+r\mathrm{/}c)-\frac{{\mu }_0}{4\pi r}n\times (n\times V(n,u))\mid \le \frac{{\mathrm{\Gamma }}_E}{r^2}\mathrm{.}}$ Subtract the same leading expression with $V$ replaced by ${\mathcal{J}}_N$. The triangle inequality, $\mathrm{\mid }n\times (n\times w)\mathrm{\mid }=\mathrm{\mid }n\times w\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }$ from [F7], and step 2.1 give ${\displaystyle \mid E(rn,u+r\mathrm{/}c)-\frac{{\mu }_0}{4\pi r}n\times (n\times {\mathcal{J}}_N(n,u))\mid \le \frac{{\mathrm{\Gamma }}_E}{r^2}+\frac{{\mu }_0}{4\pi r}\mathrm{\mid }V(n,u)-{\mathcal{J}}_N(n,u)\mathrm{\mid }\le \frac{{\mathrm{\Gamma }}_E}{r^2}+\frac{{\mu }_0{\mathcal{R}}_N(u)}{4\pi r}\mathrm{.}}$

Power remainder. The exact flux formula [F1] and step 2.1 give, pointwise in $n$, ${\displaystyle \mid \mathrm{\mid }n\times V{\mathrm{\mid }}^2-\mathrm{\mid }n\times {\mathcal{J}}_N{\mathrm{\mid }}^2\mid \le (\mathrm{\mid }n\times V\mathrm{\mid }+\mathrm{\mid }n\times {\mathcal{J}}_N\mathrm{\mid })\mathrm{\mid }n\times (V-{\mathcal{J}}_N)\mathrm{\mid }\le (\mathrm{\mid }V\mathrm{\mid }+\mathrm{\mid }{\mathcal{J}}_N\mathrm{\mid }){\mathcal{R}}_N\mathrm{.}}$ Since $\mathrm{\mid }V\mathrm{\mid }\le \mathrm{\mid }{\mathcal{J}}_N\mathrm{\mid }+{\mathcal{R}}_N$ by step 2.1, the right side is at most $(2\mathrm{\mid }{\mathcal{J}}_N\mathrm{\mid }+{\mathcal{R}}_N){\mathcal{R}}_N\le (2A_N+{\mathcal{R}}_N){\mathcal{R}}_N$. Integrating over the sphere, whose area is $4\pi$ by [F7], and multiplying by ${\mu }_0\mathrm{/}(16{\pi }^{2}c)$ gives the stated power error.

Dipole specialization. At $N=0$, ${\mathcal{J}}_0(n,u)={\int }_{B_a}{J}_t(y,u)dy=p^{\mathrm{\prime }\mathrm{\prime }}(u)$ by step 1.1 and ${\mathcal{R}}_0(u)=\frac{a}{c}{\int }_{B_a}{\sup }_{\mathrm{\mid }s-u\mathrm{\mid }\le a\mathrm{/}c}\mathrm{\mid }{J}_{tt}(y,s)\mathrm{\mid }dy$ by its definition. It is independent of $n$; the exact sphere moment [F7] therefore gives ${\displaystyle \frac{{\mu }_0}{16{\pi }^{2}c}{\int }_{S^2}\mathrm{\mid }n\times p^{\mathrm{\prime }\mathrm{\prime }}(u){\mathrm{\mid }}^{2}d\mathrm{\Omega }=\frac{{\mu }_0}{16{\pi }^{2}c}\frac{8\pi }{3}\mathrm{\mid }{p}^{\mathrm{\prime }\mathrm{\prime }}(u){\mathrm{\mid }}^2=\frac{{\mu }_0}{6\pi c}\mathrm{\mid }{p}^{\mathrm{\prime }\mathrm{\prime }}(u){\mathrm{\mid }}^2\mathrm{.}}$ Substitution in step 3.2 gives the displayed absolute dipole-power error. It gives no relative bound when the leading amplitude vanishes.