= Solution
For the forward ray of <9/b/i>, the <ultrarelativistic longitudinal Doppler factor> follows from the exact expression in terms of the <Lorentz factor>:
$$
\boxed{D=\gamma+\sqrt{\gamma^2-1}.}
$$
Indeed, $\gamma\beta=\sqrt{\gamma^2-1}$ for a positive boost. If $\beta=1-\epsilon$, the exact formulas and their consistent <asymptotic expansions> are
$$
D=\sqrt{\frac{2-\epsilon}{\epsilon}}
=\sqrt{\frac2\epsilon}\left(1-\frac\epsilon4+O(\epsilon^2)\right),
\qquad
\gamma=\frac1{\sqrt{2\epsilon-\epsilon^2}}
=\frac1{\sqrt{2\epsilon}}\left(1+\frac\epsilon4+O(\epsilon^2)\right).
$$
Equivalently,
$$
\boxed{D=2\gamma-\frac1{2\gamma}+O(\gamma^{-3})\sim2\gamma.}
$$
\b[The printed equality is not exact.] For a concrete ultrarelativistic example, $\gamma=10$ gives the correct $D=10+\sqrt{99}\simeq19.949874$, whereas $\sqrt{4\gamma^2-1}=\sqrt{399}\simeq19.974984$. The latter expands as $2\gamma-1/(4\gamma)+O(\gamma^{-3})$, with the wrong first correction.
One can obtain the printed expression by using $D^2=2/\epsilon-1$ but replacing $\gamma^2$ by only its leading approximation $1/(2\epsilon)$. This retains a constant term in the first equation while discarding a same-order term in the second. Thus $D\approx\sqrt{4\gamma^2-1}$ is admissible only as a leading relative-accuracy estimate, of the same order as $D\sim2\gamma$; it is not the stated identity or a correct next-order approximation.
Back to article page
