From the laws of conservation of the energy and the momentum, \begin{align} E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c\\ \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} = \boldsymbol{p}_e^\prime + \boldsymbol{p}_{\mathrm{ph}}^{\prime}.\\ \end{align}

Considering parallel and vertical momenta with respect to the direction of $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$,

\begin{align} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| &= p_e^\prime \cos \theta_e^\prime + p_{\mathrm{ph}}^{\prime} \cos \theta_{\mathrm{ph}}^{\prime},\\ 0 &= p_e^\prime \sin \theta_e^\prime + p_{\mathrm{ph}}^{\prime} \sin \theta_{\mathrm{ph}}^{\prime} .\\ \end{align}

From the first formula $|\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| = p_e^\prime \cos \theta_e^\prime + p_{\mathrm{ph}}^{\prime} \cos \theta_{\mathrm{ph}}^{\prime}$,

\begin{align} p_e^\prime \cos \theta_e^\prime = |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| - p_{\mathrm{ph}}^{\prime} \cos \theta_{\mathrm{ph}}^{\prime}.\\ \end{align}

By taking square of the both sides,

\begin{align} p_e^{\prime 2} \cos^2 \theta_e^\prime &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\ p_e^{\prime 2} -p_e^{\prime 2}\sin^2 \theta_e^\prime &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}.\\ \end{align} From $0 = p_e^\prime \sin \theta_e^\prime + p_{\mathrm{ph}}^{\prime} \sin \theta_{\mathrm{ph}}^{\prime}$, \begin{align} p_e^{\prime 2} -p_{\mathrm{ph}}^{\prime 2} \sin^2 \theta_{\mathrm{ph}}^{\prime} &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\ p_e^{\prime 2} &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} + p_{\mathrm{ph}}^{\prime 2} \sin^2 \theta_{\mathrm{ph}}^{\prime}-2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\ &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}. \end{align} In another way, this equation can be extracted from $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} = \boldsymbol{p}_e^\prime + \boldsymbol{p}_{\mathrm{ph}}^{\prime}$ as follows. \begin{align} \boldsymbol{p}_e^\prime &= \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} - \boldsymbol{p}_{\mathrm{ph}}^{\prime},\\ p_e^{\prime 2} &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} -2\boldsymbol{p}_{\mathrm{ph}}^{\prime}\cdot(\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}})\\ &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\ &= p_e^2 + p_{\mathrm{ph}}^2 + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\ p_e^{\prime 2} &= p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\ \end{align} where $\theta_e$ ($\theta_{\mathrm{ph}}$) is an angle between $\boldsymbol{p}_e$ ($\boldsymbol{p}_{\mathrm{ph}}$) and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$, and $\theta_{\mathrm{ph}}^{\prime}$ is an angle between $\boldsymbol{p}_{\mathrm{ph}}^{\prime}$ and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$.

Then, from the law of energy conservation $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, \begin{align} E_e^\prime = E_e + p_{\mathrm{ph}}c - p_{\mathrm{ph}}^{\prime}c.\\ \end{align} By taking square of the both sides, \begin{align} E_e^{\prime 2} = E_e^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\ \end{align} By using $E^2 = m^2c^4 + p^2c^2$, \begin{align} m_e^2c^4 + p_e^{\prime 2}c^2 &= m_e^2c^4 + p_e^{\prime 2}c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\ p_e^{\prime 2}c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\ \end{align} By using $p_e^{\prime 2} = p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}$, \begin{align} \left[p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\right]c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\ p_e p_{\mathrm{ph}} c \cos(\theta_e+\theta_{\mathrm{ph}}) - p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime} &= E_e p_{\mathrm{ph}} - E_e p_{\mathrm{ph}}^{\prime} -p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c,\\ \left[ E_e + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}\right]p_{\mathrm{ph}}^{\prime} &= \left[E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})\right]p_{\mathrm{ph}},\\ p_{\mathrm{ph}}^{\prime} &= \frac{E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})}{E_e + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}p_{\mathrm{ph}}.\\ \end{align}

If the denominator and numerator are divided by $E_e$,

\begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{1 -(p_e/E_e) \cos(\theta_e+\theta_{\mathrm{ph}})}{1 + \frac{p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}{E_e}}p_{\mathrm{ph}}.\\ \end{align}

By using $\beta_e = p_e c/E_e$,

\begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 + \frac{p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}{E_e}}p_{\mathrm{ph}}.\\ \end{align}

If unit vectors $\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}} = \boldsymbol{p}_{\mathrm{ph}}^{\prime}/p_{\mathrm{ph}}^{\prime}, \widehat{\boldsymbol{p}_{\mathrm{ph}}} = \boldsymbol{p}_{\mathrm{ph}}/p_{\mathrm{ph}}$, and $\widehat{\boldsymbol{p}_e} = \boldsymbol{p}_e/p_e$ is defined, \begin{align} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime} &= \frac{1}{p_{\mathrm{ph}}^{\prime}}p_{\mathrm{ph}}^{\prime}|\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\ &= \frac{1}{p_{\mathrm{ph}}^{\prime}}\boldsymbol{p}_{\mathrm{ph}}^{\prime}\cdot(\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}})\\ &= \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot(\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}})\\ &= \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\boldsymbol{p}_e + \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\boldsymbol{p}_{\mathrm{ph}}\\ &= p_e\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + p_{\mathrm{ph}}\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}}.\\ \end{align}

By using this formula,

\begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 + \frac{p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}{E_e}}p_{\mathrm{ph}}\\ &= \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 + \frac{p_{\mathrm{ph}} c - p_e c\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} - p_{\mathrm{ph}}c\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}}}{E_e}}p_{\mathrm{ph}}\\ &= \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c}{E_e} (1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}p_{\mathrm{ph}}.\\ \end{align}

Finally,

\begin{align} p_{\mathrm{ph}}^{\prime} = \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c}{E_e} (1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}p_{\mathrm{ph}}.\\ \end{align}

On the SLEGS paper (Nucl. Instrum. Phys. Meth. Res. A 1033, 1666742 (2022)), there is a formula to calculate the energy of produced $\gamma$ ray on the slant scattering as follows. \begin{align} E_\gamma = \frac{E_l(1 - \beta \cos\theta)}{1 - \beta \cos\theta_\gamma + \frac{E_l(1 - \cos\eta)}{E_e}},\\ \end{align} where $E_\gamma$ ($E_l$) is energy of the scattered photon (incident lasesr photon), $\beta$ is ratio of speed of the incident electron to that of light, $\theta$ ($\theta_\gamma$) is angle between the inicident electron direction and incident laser photon (scattered photon), and $\eta$ is angle between the incident laser photon and scattered photon. The definition of the angles can be seen in Fig. 1 of the paper.

From the above general formula,

\begin{align} p_{\mathrm{ph}}^{\prime} = \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c}{E_e} (1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}p_{\mathrm{ph}},\\ p_{\mathrm{ph}}^{\prime}c = \frac{p_{\mathrm{ph}}c\left[1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})\right]}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c(1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}{E_e}}.\\ \end{align} According to definitions of the slant scattering paper, \begin{align} E_\gamma &= p_{\mathrm{ph}}^{\prime}c,\\ E_l &= p_{\mathrm{ph}}c,\\ \theta &= \theta_e+\theta_{\mathrm{ph}},\\ \beta &= \beta_e,\\ \cos\theta_\gamma &= \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e},\\ \cos\eta &= \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}}.\\ \end{align} Then, one can get \begin{align} E_\gamma = \frac{E_l(1 - \beta \cos\theta)}{1 - \beta \cos\theta_\gamma + \frac{E_l(1 - \cos\eta)}{E_e}}.\\ \end{align}

Considering the most simple case: head-on scattering

From the laws of conservation of the energy and the momentum, \begin{align} E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c\\ p_e - p_{\mathrm{ph}} = p_e^\prime + p_{\mathrm{ph}}^{\prime},\\ \end{align} where $E_e$ ($p_e$) is the energy (momentum) of the electron before the scattering, $E_e^\prime$ ($p_e^\prime$) is the energy (momentum) of the electron after the scattering, and $p_{\mathrm{ph}}$ ($p_{\mathrm{ph}}^{\prime}$) is the momentum of photon before the scattering (after the scattering).

By eliminating $p_{\mathrm{ph}}^{\prime}$ from the two equations, \begin{align} E_e + 2p_{\mathrm{ph}}c - p_{e}c = E_e^\prime - p_e^{\prime}c\\ E_e + 2p_{\mathrm{ph}}c - p_{e}c + p_e^{\prime}c = \sqrt{p_e^{{\prime}2}c^2+m_{e}^2c^4}.\\ \end{align} By making the square of both sides, \begin{align} \left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)^2 + p_e^{{\prime}2}c^2 + 2p_e^{{\prime}}c\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) = p_e^{{\prime}2}c^2+m_{e}^2c^4.\\ \end{align} Then, \begin{align} 2p_e^{{\prime}}c\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) = m_{e}^2c^4 -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)^2.\\ p_e^{{\prime}} = \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right].\\ \end{align}

From $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, \begin{align} E_e^\prime = E_e + p_{\mathrm{ph}}c - p_{\mathrm{ph}}^{\prime}c.\\ \end{align} By taking square of the both sides, \begin{align} E_e^{\prime 2} = E_e^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\ \end{align} By using $E^2 = m^2c^4 + p^2c^2$,

\begin{align} m_e^2c^4 + p_e^{\prime 2}c^2= m_e^2c^4 + p_e^{\prime 2}c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\ p_e^{\prime 2}c^2= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\ \end{align} On the other hand, from $p_e - p_{\mathrm{ph}} = p_e^\prime + p_{\mathrm{ph}}^{\prime}$, \begin{align} p_e^\prime = p_e - p_{\mathrm{ph}} - p_{\mathrm{ph}}^{\prime}. \end{align} By taking square of the both sides, \begin{align} p_e^{\prime 2} = p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} - 2 p_e p_{\mathrm{ph}} - 2 p_e p_{\mathrm{ph}}^{\prime} + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}. \end{align} Therefore, by using $p_e^{\prime 2}c^2= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2$, \begin{align} (p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} - 2 p_e p_{\mathrm{ph}} - 2 p_e p_{\mathrm{ph}}^{\prime} + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime})c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\ - 2 p_e p_{\mathrm{ph}}c^2 - 2 p_e p_{\mathrm{ph}}^{\prime}c^2 + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}c^2 &= 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\ E_e p_{\mathrm{ph}}^{\prime} - p_e p_{\mathrm{ph}}^{\prime}c + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}c &= E_e p_{\mathrm{ph}} + p_e p_{\mathrm{ph}}c,\\ \left[ E_e - p_e c + 2 p_{\mathrm{ph}}c \right]p_{\mathrm{ph}}^{\prime} &= \left[E_e + p_e c\right]p_{\mathrm{ph}},\\ p_{\mathrm{ph}}^{\prime} &=\frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\ \end{align} By using $E_e = m_e \gamma_ec^2,\ p_e = m_e \gamma_e\beta_e c,\ \beta_e = p_e c/E_e$, \begin{align} p_{\mathrm{ph}}^{\prime} &=\frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\ &=\frac{1 + p_{e}c / E_e}{1-p_{e}c / E_e + 2p_{\mathrm{ph}}c/E_e}p_{\mathrm{ph}}\\ &=\frac{1 + \beta_e}{1-\beta_e + 2p_{\mathrm{ph}}c/E_e}p_{\mathrm{ph}} \end{align} Finally, \begin{align} p_{\mathrm{ph}}^{\prime} &=\frac{1 + \beta_e}{1-\beta_e + 2p_{\mathrm{ph}}c/(m_e \gamma_ec^2)}p_{\mathrm{ph}}\\ \end{align}

By using the above $E_{e}^{\prime}$, \begin{align} p_{\mathrm{ph}}^{\prime} &= p_e - p_{\mathrm{ph}} -p_e^\prime \\ &= p_e - p_{\mathrm{ph}} - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right] \\ &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) -2c\left(p_e - p_{\mathrm{ph}}\right) \right] \\ &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + p_{e}c\right) \right] \\ &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 -\left(E_e + p_{e}c\right)\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\ &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-\left(E_e^2 - p_{e}^2c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\ &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\ &= - \frac{1}{2c}\left[\frac{-2p_{\mathrm{ph}}\left(E_e + p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\ &= p_{\mathrm{ph}}\frac{E_e + p_{e}c}{E_e + 2p_{\mathrm{ph}}c - p_{e}c}\\ \end{align} Then, \begin{align} p_{\mathrm{ph}}^{\prime} = \frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\ \end{align}

From the general formula, \begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})}{E_e + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}p_{\mathrm{ph}}.\\ \end{align} For head-on scattering, \begin{align} \theta_e+\theta_{\mathrm{ph}} &= \pi,\\ |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| &= p_e - p_{\mathrm{ph}},\\ \theta_\mathrm{ph}^{\prime} &= 0.\\ \end{align} Then, \begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{E_e +p_e c}{E_e + p_{\mathrm{ph}} c - (p_e - p_{\mathrm{ph}}) c}p_{\mathrm{ph}}.\\ \end{align} Finally, \begin{align} p_{\mathrm{ph}}^{\prime} &= \frac{E_e +p_e c}{E_e - p_e c + 2p_{\mathrm{ph}} c}p_{\mathrm{ph}}.\\ \end{align}