Photon energy
$$E_{\gamma}=h\nu_{\gamma}=\frac{hc}{\lambda_{\gamma}}$$
and momentum
$$p_{\gamma}=\frac{h\nu_{\gamma}}{c}=\frac{h}{\lambda_{\gamma}}$$

The photon scatters on the electron:
$$p_{\gamma}c+m_e c^2=p_{\gamma'}c+\sqrt{(m_e c^2)^2+(p_e c)^2}$$
$$\vec{p}_{\gamma}=\vec{p}_{\gamma'}+\vec{p}_e$$

Increase in wavelength of the photon
$$\lambda_{\gamma'}-\lambda_{\gamma}=\frac{h}{m_e c}\left(1-\cos(\theta)\right)$$