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For a localized charge and current distribution, take the retarded vector potentialLet $R=|x|$, $n=x/R$, and $t_R=t-R/c$. If the source size is much smaller than both $R$ and the characteristic radiation wavelength, the leading dipole term iswhere the last identity follows from charge conservation.

Keeping only the $1/R$ terms in the radiation zone gives the transverse fieldsThe radial Poynting vector therefore yieldsUsingor equivalently $\int {\sin }^2\theta d\Omega =8\pi /3$, we recover the electric dipole radiation formulaThis requires a localized, nonrelativistic source of size $d\ll \lambda$, observation distance $R\gg d$ and $R\gg \lambda$, and neglect of higher multipoles and faster-decaying near fields.

For the pulsar, write its magnetic dipole moment as $m(t)$ to distinguish it from the electric dipole above. The magnetic dipole radiation formula differs by a factor $c^{-2}$:Hereand hence

On the slow spin-down timescale, energy conservation gives the magnetic-dipole spin-down equationsoHalf of the initial rotational energy remains when $\Omega ={\Omega }_0/\sqrt{2}$. Thereforewhere $I=2M{\mathcal{R}}^2/5$. Since $E_0=I{\Omega }_0^2/2$, the equivalent expression is

Runge's polynomial approximation theorem says that if $K\subset \mathbb{C}$ is compact, $\mathbb{C}\setminus K$ is connected, and $f$ is holomorphic on a neighborhood of $K$, then for every $\epsilon >0$ there is a polynomial $p$ such that

LetThis arc is compact, its complement is connected, and $1/z$ is holomorphic on a neighborhood of it. Applying Runge with error $1/n$ gives a polynomial $q_n$ satisfyingwhich is the requested uniform approximation.

For the pointwise construction, for $n\ge 2$ setThe three pieces are disjoint compact sets and $\mathbb{C}\setminus K_n$ is connected. Define a function on a neighborhood of $K_n$ to equal $1$, $-1$, and $0$ on neighborhoods of $K_n^{+}$, $K_n^{-}$, and $K_n^0$, respectively. It is holomorphic because those neighborhoods may be chosen disjoint. Runge supplies a polynomial $P_n$ such thatEvery fixed point with $|z|<1$ eventually belongs to the appropriate one of these three sets, according to the sign of its real part. These inequalities therefore give precisely the asserted pointwise limits.