We carry out a detailed study of the computation of external Compton Spectrum due to scattering between relativistic electrons with the seed photons emitted by any source. The analysis is done considering the general scenario by taking care of all possible relativistic terms, making the model applicable across the ranges of particle Lorentz factors and seed photon energies as well as their anisotropic nature. This analysis is an extension of Blumenthal-Gould's seminal work which works from non-relativistic to relativistic gas and low energy to high energy photons. The application of the work is discussed for a couple of cases 1. Scattered spectrum from particles in above Pulsar atmosphere and 2. Compton spectrum produced by Maxwell Juttner gas (or Maxwellian Plasma) in AGN atmosphere or around microquasars, which is typically useful in explaining the soft state spectra of such systems.