  Our main objective is to compare the analytic properties of hydrodynamic
series with the stability and causality conditions applied to hydrodynamic
modes. Analyticity, in this context, implies that the hydrodynamic series
behaves as a univalent or single-valued function. Stability and causality
adhere to physical constraints where hydrodynamic modes neither exhibit
exponential growth nor travel faster than the speed of light. Through an
examination of various hydrodynamic models, such as the Muller-Israel-Stewart
(MIS) and the first-order hydro models like the BDNK
(Bemfica-Disconzi-Noronha-Kovtun) model, we observe no new restrictions
stemming from the analyticity limits in the shear channel of these models.
However, local univalence is maintained in the sound channel of these models
despite the global divergence of the hydrodynamic series. Notably, differences
in the sound equations between the MIS and BDNK models lead to distinct
analyticity limits. The MIS model's sound mode remains univalent at high
momenta within a specific transport range. Conversely, in the BDNK model, the
univalence of the sound mode extends to intermediate momenta across all stable
and causal regions. Generally, the convergence radius is independent of
univalence and the given dispersion relation predominantly influences their
correlation. For second-order frequency dispersions, the relationship is
precise, i.e. within the convergence radius, the hydro series demonstrates
univalence. However, with higher-order dispersions, the hydro series is locally
univalent within certain transport regions, which may fall within or outside
the stable and causal zones.
