= Ultrarelativistic longitudinal Doppler factor
{title2=$D=\gamma+\sqrt{\gamma^2-1}=2\gamma-1/(2\gamma)+O(\gamma^{-3})$}
For a forward <photon> and a positive collinear <Lorentz transformation>, the <Doppler factor> is $D=\gamma(1+\beta)$. The identity $\gamma\beta=\sqrt{\gamma^2-1}$ gives the exact expression in the title. Expanding the square root by the <binomial series> gives $D=2\gamma-1/(2\gamma)-1/(8\gamma^3)+O(\gamma^{-5})$. A leading approximation $D\sim2\gamma$ has relative error of order $\gamma^{-2}$; retaining a purported subleading term requires expanding all dependent quantities consistently. For example, $\sqrt{4\gamma^2-1}$ has subleading term $-1/(4\gamma)$ and is not the exact longitudinal <Doppler factor>.
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