Complex analytical structure of Stokes wave for two-dimensional
potential flow of the ideal incompressible fluid with free surface
and infinite depth is analyzed. Stokes wave is the fully nonlinear
periodic gravity wave propagating with the constant velocity. We
consider Stokes wave in the conformal variables which maps the
domain occupied by fluid into the lower complex half-plane. Then
Stokes wave can be described through the position and the type of
complex singularities in the upper complex half-plane. Similar idea
was exploited for other hydrodynamic systems in different
approximations. We studied fully nonlinear problem and identified
that this singularity is the square-root branch point. That branch
cut defines the second sheet of the Riemann surface if we cross the
branch cut. Second singularity is also the square-root and located
in that second (nonphysical) sheet of the Riemann surface in the
lower half-plane. Crossing corresponding branch cut in second sheet
one arrives to the third sheet of Riemann surface with another
singularity etc. As the nonlinearity increases, all singularities
approach the real line forming the classical Stokes solution
(limiting Stokes wave) with the branch point of power 2/3. We
reformulated Stokes wave equation through the integral over jump at
the branch cut which provides the efficient way for finding of the
explicit form of Stokes wave.