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We perform a perturbative calculation of the third order optical conductivities of doped graphene, using approximations valid around the Dirac points and neglecting effects due to scattering and electron–electron interactions. In this limit analytic formulas can be constructed for the conductivities. We discuss in detail the results for third harmonic generation, the Kerr effect and two-photon carrier injection, parametric frequency conversion, and two-color coherent current injection. We find a complicated dependence on the chemical potential and photon energies. The linear dispersion causes resonances over a wide range of photon energies, and it is possible to obtain large optical nonlinearities by tuning the chemical potential.