Analyticity of the Free Surface for Periodic Travelling Capillary-Gravity Water Waves with Vorticity

For capillary-gravity water waves with vorticity, in the absence of stagnation points, we prove that if the vorticity function has a Hölder continuous derivative then the free surface will be smooth. Furthermore, once we assume that the vorticity function is real analytic it will follow that the wave surface profile is also analytic. In particular this scenario includes the case of irrotational fluid flow where the vorticity is zero.