Ultrarelativistic longitudinal Doppler factor ($D=\gamma +\sqrt{{\gamma }^2-1}=2\gamma -1/(2\gamma )+O({\gamma }^{-3})$)

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 $D\sim 2\gamma$ 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.