
$$\eqalign{
 \lambda_f   &=  \left[ \lambda^{\prime}_i + {{2h}\over{m_e c}} \right]
                          \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  \cr
} $$


\bigskip

$$\eqalign{
 \lambda_f   &=  \lambda^{\prime}_f \sqrt{ {(1 - v/c)}\over{(1 + v/c)}}  \cr
             &\phantom{=} \cr
             &=  \left[ \lambda^{\prime}_i + {{2h}\over{m_e c}} \right]
                          \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  \cr
             &\phantom{=} \cr
             &=  \left[ \left(\lambda_i \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} } \right)
                                                     + {{2h}\over{m_e c}} \right] 
                          \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  \cr
} $$


\bigskip

$$
\lambda^{\prime}_i \gg {{2 h}\over{m_e c}}
$$

\bigskip

$$
\lambda^{\prime}_i + {{2 h}\over{m_e c}} \simeq \lambda^{\prime}_i 
$$


\bigskip

$$
\left[ \lambda^{\prime}_i + {{2h}\over{m_e c}} \right]  \simeq \lambda^{\prime}_i 
$$

$$\eqalign{
\lambda_f &\simeq  \lambda^{\prime}_i 
           \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  \cr
             &\phantom{=} \cr
          &\simeq  \lambda_i 
           \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  
           \sqrt{ {{(1 - v/c)}\over{(1 + v/c)}} }  \cr
             &\phantom{=} \cr
          &\simeq  \lambda_i 
             \left[ {{(1 - v/c)}\over{(1 + v/c)}} \right]  \cr
             &\phantom{=} \cr
          &\simeq  \lambda_i 
             \left[ {{(1 - v/c)}\over{2}} \right]  \cr
}$$

\bigskip



\bigskip

$$\eqalign{
  { 1 - {{v}\over{c}}} &\simeq 2 \left( {{\lambda_f}\over{\lambda_i}} \right) \cr
             &\phantom{=} \cr
        {{v}\over{c}} &\simeq 1 - 2 \left( {{\lambda_f}\over{\lambda_i}} \right) \cr
}$$

\bye
