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Physical interpretation

A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function. Suppose that the pair of (twice continuously differentiable) functions u and v satisfies the Cauchy–Riemann equations. We will take u to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the velocity vector of the fluid at each point of the plane is equal to the gradient of u, defined by ∇ u = ∂ u ∂ x i + ∂ u ∂ y j. {\displaystyle \nabla u={\frac {\partial u}{\partial x}}\mathbf {i} +{\frac {\partial u}{\partial y}}\mathbf {j}.}

By differentiating the Cauchy–Riemann equations for the functions u and v, with the symmetry of second derivatives, one shows that u solves Laplace's equation: ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0. {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.} That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible.

The function v also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the dot product ∇ u ⋅ ∇ v = 0 {\textstyle \nabla u\cdot \nabla v=0} (∇ u ⋅ ∇ v = ∂ u ∂ x ⋅ ∂ v ∂ x + ∂ u ∂ y ⋅ ∂ v ∂ y = ∂ u ∂ x ⋅ ∂ v ∂ x − ∂ u ∂ x ⋅ ∂ v ∂ x = 0 {\textstyle \nabla u\cdot \nabla v={\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\cdot {\frac {\partial v}{\partial y}}={\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}=0}), i.e., the direction of the maximum slope of u and that of v are orthogonal to each other. This implies that the gradient of u must point along the v = const {\textstyle v={\text{const}}} curves; so these are the streamlines of the flow. The u = const {\textstyle u={\text{const}}} curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves u = const {\textstyle u={\text{const}}} and v = const {\textstyle v={\text{const}}}. Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal family of curves. At the points where ∇ u = 0 {\textstyle \nabla u=0}, the stationary points of the flow, the equipotential curves of u = const {\textstyle u={\text{const}}} intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.