Derivaton of formula for caculation of the wavelength shift or compton Equation

For a compton scattering, the following assumptions were made

1.      the collision between the high energy photon  and  the electron  is elastic. the energy and the momentum carried by the scattering photon and recoil electron is governed by the law of conservation of energy  and momentum respectively.

2.      an x-ray of frequency  f has energy  hf and momentum hf/c

3.      the electrons are comparatively at rest  in respect to the x-ray.

the above  equations shows that the increase in wavelength is independent of the wavelength of the incident radiations as well as the nature of the scattering but depend on the angle of the scattering