  Hydrodynamic excitations corresponding to sound and shear modes in fluids are
characterised by gapless dispersion relations. In the hydrodynamic gradient
expansion, their frequencies are represented by power series in spatial
momenta. We investigate the analytic structure and convergence properties of
the hydrodynamic series by studying the associated spectral curve in the space
of complexified frequency and complexified spatial momentum. For the strongly
coupled ${\cal N}=4$ supersymmetric Yang-Mills plasma, we use the holographic
duality methods to demonstrate that the derivative expansions have finite
non-zero radii of convergence. Obstruction to the convergence of hydrodynamic
series arises from level-crossings in the quasinormal spectrum at complex
momenta.
