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Thin Airfoil Flow Equation (Cauchy-Riemann)

The governing equation for the complex potential W(z)W(z) in 2D flow, often simplified to W(z)=U∞z+12πi∫CΓ(z′)z−z′dz′W(z) = U_{\infty} z + \frac{1}{2\pi i} \int_C \frac{\Gamma(z')}{z-z'} dz'.
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The statement of the theorem

In the complex plane z=x+iyz = x + iy, the complex potential W(z)W(z) for the 2D flow is defined. The governing equation relating W(z)W(z) to the circulation Γ\Gamma and the freestream velocity U∞U_{\infty} is: W(z)=U∞z+12πi∫CΓ(z′)z−z′dz′W(z) = U_{\infty} z + \frac{1}{2\pi i} \int_C \frac{\Gamma(z')}{z-z'} dz' where CC is the contour enclosing the airfoil.
Source: Wikipedia