Three-dimensional analog of the Sokhotsky–Plemelj formulas and its application in the wing theory

A solution of a hydrodynamic problem of motion of an ideal incompressible fluid in a finite-thickness vortex layer is obtained. In the limiting case (infinitely thin layer), this layer transforms to a vortex surface. Formulas are derived for limiting values of the velocity vector of the fluid approaching this surface; these formulas extend the Sokhotsky–Plemelj formulas for a singular integral of the Cauchy type to a three-dimensional space. Three integral equations are derived on the basis of these formulas and the proposed method of modeling a finite-thickness wing by a closed vortex surface. It is shown that only one equation is left in the case of an infinitely thin wing, which corresponds to the condition of fluid non-penetration through the wing surface.