Cauchy-Riemann conditions and vector fields

Question

Can Cauchy-Riemann conditions of Complex Functions valid for vector fields, I observe when vector fields are irrotational and incompressible they possess a result similar to Cauchy-Riemann conditions. Does this imply that every analytic function us divergentless and irrotational?

Answer 1

When a vector field $\mathbf{u}$ is irrotational ($\nabla \times \mathbf{u}=0$) there is a potential $\phi$ such that $\nabla \phi = \mathbf{u}$. The condition for incompressibility $\nabla \cdot \mathbf{u}=0$ leads then to $\nabla^2 \phi = 0$, i.e., the potential of an irrotational and incompressible vector field is a solution of the Laplace equation (i.e., it's a harmonic equation). Also, any complex analytical (or holomorphic) function is also a solution of the Laplace equation. Therefore, one can say that the Cauchy-Riemann conditions for complex functions and the condition of incompressibility and irrotationality of a vector field are two faces of the same coin.

Answer 2

Yes, if you split a complex analytic (i.e. holomorphic) function $$f(z) = u(x, y) + i\, v(x, y) \, \text{ where } \, z = x + i\, y \in \mathbb{C}$$ into real part $\text{ Re}f(z) = u(x,y)$ and imaginary part $\text{ Im}f(z) = v(x,y)$, and form the vector field $$\vec{V}(x,y) = v(x, y) \, \frac{\partial}{\partial x} \, - \, u(x, y) \, \frac{\partial}{\partial y}$$ then that vector field $\vec{V}(x,y)$ has zero curl and zero divergence: $$\nabla \times \vec{V} = \vec{0} \,\text{ and } \, \nabla \circ \vec{V} = 0$$ A vector field with the latter two properties, zero curce and zero divergence, is called a harmonic vector field.

And vice versa: if you have a 2D harmonic vector field $$\vec{V}(x,y) = a(x, y) \frac{\partial}{\partial x} + b(x, y) \frac{\partial}{\partial y}$$ i.e. $\vec{V}$ has zero curl and zero divergence, then its components are the real and the imaginary part of a holomorphic function: more precisely, the function $$f(z) = b(x, y) - i\, a(x, y) \, \text{ where } \, z = x + i\, y \in \mathbb{C}$$ is a complex analytic (i.e. holomorphic) function.

Proof: Cauchy-Riemann equations.