  We develop the plasmon-pole approximation (PPA) theory for calculating the
carrier self-energy of extrinsic graphene as a function of doping density
within analytical approximations to the $GW$ random phase approximation
($GW$-RPA). Our calculated self-energy shows excellent quantitative agreement
with the corresponding full $GW$-RPA calculation results in spite of the
simplicity of the PPA, establishing the general validity of the plasmon-pole
approximation scheme. We also provide a comparison between the PPA and the
hydrodynamic approximation in graphene, and comment on the experimental
implications of our findings.
