  Weyl semimetal (WSM) feature tilted Dirac cones and can be type I or II
depending on the magnitude of the tilt parameter ($C$). The boundary between
the two types is at $C=1$ where the cones are tipped and there is a Lifshitz
transition. The topological charge of a WSM is one. In multi-Weyl it can be two
or more depending on the value of the winding number $J$. We calculate the
absorptive part of the AC optical conductivity both along the tilt direction
($\sigma_{zz}$) and perpendicular to it ($\sigma_{xx}$) as a function of the
tilt ($C$) and chemical potential ($\mu$). For zero tilt there is a
discontinuous rise in both $\sigma_{xx}$ and $\sigma_{zz}$ at photon energy
$\Omega=2\mu$ followed by the usual linear in $\Omega$ law for $\sigma_{xx}$ at
$J=1,2$ and $\sigma_{zz}$ at $J=1$. For $J=2$ and $\sigma_{zz}$ the interband
background is constant rather than linear in $\Omega$. For type I there is a
readjustment of optical spectral weight as the tilt is increased. The
absorption starts from zero at $2\mu/(1+C)$ and then rises in a quasilinear
fashion till it merge with the usual undoped untilted interband background at
$2\mu/(1-C)$. The discontinuous rise at twice the chemical potential of the
untilted case is lost. For type II the interband background of the undoped
untilted case is never recovered. For noncentrosymmetric materials the energies
of a pair of opposite chirality Weyl nodes become shifted by $\pm Q_{0}$ and
this leads to two separate absorption edges corresponding to the effective
chemical potential of each of the two nodes at $2(\mu+\chi Q_{0})$ depending on
chirality $\chi=\pm$. We provide analytic expressions for the conductivity in
this case which depend only on the ratio $Q_{0}/\mu$ and tilt when plotted
against $\Omega/\mu$. The signature of finite energy shift $Q_{0}$ is more
pronounced for $\sigma_{zz}$ and $J=2$ than for the other cases.
