  Under appropriate conditions for the initial configuration, the empirical
measure of the $N$-particle Dyson model with parameter $\beta \geq 1$ converges
to a unique measure-valued process as $N$ goes to infinity, which is
independent of $\beta$. The limit process is characterized by its Stieltjes
transform called the Green's function. Since the Green's function satisfies the
complex Burgers equation in the inviscid limit, this is called the hydrodynamic
limit of the Dyson model. We review the relations among the hydrodynamic
equation of the Green's function, the continuity equation of the probability
density function, and the functional equation of the Green's function. The
basic tools to prove the relations are the Hilbert transform, a special case of
the Sokhotski-Plemelj theorem, and the method of characteristics for solving
partial differential equations. For two special initial configurations, we
demonstrate how to characterize the limit processes using these relations.
