  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. Simulations with the quadruple and variable
precisions are performed to find Stokes wave with high accuracy and study the
Stokes wave approaching its limiting form with $2\pi/3$ radians angle on the
crest. A conformal map is used which maps a free fluid surface of Stokes wave
into the real line with fluid domain mapped into the lower complex half-plane.
The Stokes wave is fully characterized by the complex singularities in the
upper complex half-plane. These singularities are addressed by rational
(Pad\'e) interpolation of Stokes wave in the complex plane. Convergence of
Pad\'e approximation to the density of complex poles with the increase of the
numerical precision and subsequent increase of the number of approximating
poles reveals that the only singularities of Stokes wave are branch points
connected by branch cuts. The converging densities are the jumps across the
branch cuts. There is one branch cut per horizontal spatial period $\lambda$ of
Stokes wave. Each branch cut extends strictly vertically above the
corresponding crest of Stokes wave up to complex infinity. The lower end of
branch cut is the square-root branch point located at the distance $v_c$ from
the real line corresponding to the fluid surface in conformal variables. The
limiting Stokes wave emerges as the singularity reaches the fluid surface.
Tables of Pad\'e approximation for Stokes waves of different heights are
provided. These tables allow to recover the Stokes wave with the relative
accuracy of at least $10^{-26}$. The tables use from several poles to about
hundred poles for highly nonlinear Stokes wave with $v_c/\lambda\sim 10^{-6}.$
