λ_{f} = [λ′_{i} + \frac{2h}{m_{e}c}] \sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

λ_{f} = λ′_{f} \sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

= [λ′_{i} + \frac{2h}{m_{e}c}] \sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

= [(λ_{i}\sqrt{\frac{(1 − v/c)}{(1 + v/c)}}) + \frac{2h}{m_{e}c}] \sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

λ′_{i} ≫ \frac{2h}{m_{e}c}

λ′_{i} + \frac{2h}{m_{e}c} ≃ λ′_{i}

[λ′_{i} + \frac{2h}{m_{e}c}] ≃ λ′_{i}

λ_{f} ≃ λ′_{i}\sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

≃ λ_{i}\sqrt{\frac{(1 − v/c)}{(1 + v/c)}}\sqrt{\frac{(1 − v/c)}{(1 + v/c)}}

≃ λ_{i} [\frac{(1 − v/c)}{(1 + v/c)}]

≃ λ_{i} [\frac{(1 − v/c)}{2}]

\begin{aligned}1 − \frac{v}{c} & ≃ 2 (\frac{λ_{f}}{λ_{i}} ) \\ \frac{v}{c} & ≃ 1 − 2 (\frac{λ_{f}}{λ_{i}} ) \end{aligned}