Manifestation of the electric dipole moment in the decays of $\tau$ leptons produced in $e^{+}{e}^{-}$ annihilation

$CP$-odd asymmetries in the processes $e^{+}{e}^{-}\to {\tau }^{+}{\pi }^{-}{\nu }_{\tau }$, $e^{+}{e}^{-}\to {\pi }^{+}{\tau }^{-}{\bar{\nu }}_{\tau }$, $e^{+}{e}^{-}\to {\tau }^{+}{\rho }^{-}{\nu }_{\tau }$, $e^{+}{e}^{-}\to {\rho }^{+}{\tau }^{-}{\bar{\nu }}_{\tau }$, $e^{+}{e}^{-}\to {\tau }^{+}{e}^{-}{\nu }_{\tau }{\bar{\nu }}_e,$ and $e^{+}{e}^{-}\to {\tau }^{-}{e}^{+}{\nu }_e{\bar{\nu }}_{\tau }$ are investigated with account for longitudinal polarization of electron (or positron) beam. These asymmetries is a manifestation of electric dipole form factor $F_3^{\tau }\equiv b$ in the $\gamma {\tau }^{+}{\tau }^{-}$ vertex. It is shown that, to measure $Imb$ in the specified processes, polarization is not needed, while to measure $Reb$ it is required. The processes $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\nu }_{\tau }{\bar{\nu }}_{\tau }$, $e^{+}{e}^{-}\to e^{+}{e}^{-}{\nu }_{\tau }{\bar{\nu }}_{\tau }{\nu }_e{\bar{\nu }}_e$, $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}{\nu }_{\tau }{\bar{\nu }}_{\tau }{\nu }_{\mu }{\bar{\nu }}_{\mu }$, $e^{+}{e}^{-}\to {\mu }^{+}{e}^{-}{\nu }_{\tau }{\bar{\nu }}_{\tau }{\nu }_{\mu }{\bar{\nu }}_e$, and $e^{+}{e}^{-}\to {\mu }^{-}{e}^{+}{\nu }_{\tau }{\bar{\nu }}_{\tau }{\nu }_e{\bar{\nu }}_{\mu }$ are also discussed for the case of unpolarized electron and positron beams. In the latter cases it is possible to measure $Reb$ using the differential cross section over momenta of both registered particles.