A photon with initial energy E_{gamma _ i} = h
u _ i travelling in the hat{x} direction scatters of an electron at rest (E_{e_{i}} = m_ ec^2). After the scatter, the photon has energy E_{gamma _ f} = h
u _ f and is travelling at an angle θ relative to the original hat{x} direction. The electron has energy given by E_{e_ f} = sqrt { p_ e^2 c^2 + m_ e^2 c^4} and has scattered at a different angle. See figure below for a depiction of the relevant variables.
The conservation of energy tells us