Derivation of a Relativistic Compton Wave

In 1923, Arthur Holly Compton introduced what today is known as the Compton wave. Even if the Compton scattering derivation by Compton is relativistic in the sense that it takes into account the momentum of photons traveling at the speed of light, the original Compton derivation indirectly assumes that the electron is stationary at the moment it is scattered by electrons, but not after it has been hit by photons. Here, we extend this to derive Compton scattering for the case when the electron is initially moving at a velocity v. This gives us a relativistic Compton wave, something we remarkably have not seen published before.