## The Cauchy - Riemann Equations

The Cauchy – Riemann equations lay down the conditions for a complex valued function to be differentiable at a point, but they also have applications in fluid dynamics. If then is differentiable at a point if: Let the functions and be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The equations are occur in fluid mechanics from consideration of the velocity potential  and are found from the velocity potential  The condition that the flow be incompressible is that and the condition that the flow be irrotational is that If we define the differential of a function by then the incompressibility condition is the integrability condition for this differential: the resulting function is actually the stream function because it is constant along flow lines. The first derivatives of are given by and the irrotationality condition implies that satisfies the Laplace equation: The harmonic function (harmonic since ) that is conjugate to is the velocity potential.

Every analytic function corresponds to a steady incompressible, irrotational fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function. 