  We obtain analytic expressions for the conductivity of pristine (pure)
graphene in the framework of the Dirac model using the polarization tensor in
(2+1)-dimensions defined along the real frequency axis. It is found that at
both zero and nonzero temperature $T$ the in-plane and out-of-plane
conductivities of graphene are equal to each other with a high precision and
essentially do not depend on the wave vector. At $T=0$ the conductivity of
graphene is real and equal to $\sigma_0=e^2/(4\hbar)$ up to small nonlocal
corrections in accordance with many authors. At some fixed $T\neq 0$ the real
part of the conductivity varies between zero at low frequencies $\omega$ and
$\sigma_0$ for optical $\omega$. If $\omega$ is fixed, the conductivity varies
between $\sigma_0$ at low $T$ and zero at high $T$. The imaginary part of the
conductivity of graphene is shown to depend on the ratio of $\omega$ to $T$. In
accordance to the obtained asymptotic expressions, at fixed $T$ it varies from
infinity at $\omega=0$ to a negative minimum value reached at some $\omega$,
and then approaches to zero with further increase of $\omega$. At fixed
$\omega$ the imaginary part of the conductivity varies from zero at $T=0$,
reaches a negative minimum at some $T$ and then goes to infinity together with
$T$. The numerical computations of both the real and imaginary parts of the
conductivity are performed. The above results are obtained in the framework of
quantum electrodynamics at nonzero temperature and can be generalized for
graphene samples with nonzero mass gap parameter and chemical potential.
