Question

Let \(f(z)=u+i v\) be an analytic function, and let \(\mathbf{F}\) be the vector \(\mathbf{F}=v \mathbf{i}+u \mathbf{j}\). Show that the equations div \(\mathbf{F}=0\) and \(\operatorname{curl} \mathbf{F}=0\) are equivalent to the Cauchy-Riemann equations.

Short answer

The equations \(\operatorname{div} \mathbf{F}=0\) and \(\operatorname{curl} \mathbf{F}=0\) are equivalent to the Cauchy-Riemann equations.

Step-by-step solution

Define the Terms

Let an analytic function be defined as follows: \[f(z) = u(x,y) + i v(x,y),\] where \(u\) and \(v\) are real-valued functions of \(x\) and \(y\). The vector \(\mathbf{F}\) is given by \(\mathbf{F} = v \mathbf{i} + u \mathbf{j}\).

Compute Divergence of \(\mathbf{F}\)

The divergence of a vector field \(\mathbf{F} = v \mathbf{i} + u \mathbf{j}\) in two dimensions is given by:\[\operatorname{div} \mathbf{F} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}.\] For \(\operatorname{div} \mathbf{F} = 0\) we need:\[\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} = 0.\]

Compute Curl of \(\mathbf{F}\)

The curl of a vector field in two dimensions, where \(\mathbf{F} = v \mathbf{i} + u \mathbf{j}\), is given by:\[\operatorname{curl} \mathbf{F} = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}.\] For \(\operatorname{curl} \mathbf{F} = 0\) we need:\[\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} = 0.\]

State the Cauchy-Riemann Equations

The Cauchy-Riemann equations for an analytic function \(f(z)=u(x,y) + iv(x,y)\) are:\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.\]

Show Equivalence

Comparing the results from Step 2 and Step 3 with the Cauchy-Riemann equations, we can see that:\[\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} = 0 \quad \iff \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},\] and \[\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} = 0 \quad \iff \quad \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}.\] Hence, the divergence and curl conditions are satisfied if and only if the Cauchy-Riemann equations are satisfied.

Key concepts

analytic functions

In complex analysis, an analytic function (also known as a holomorphic function) is a function that is locally given by a convergent power series. For a complex function to be analytic, it must be differentiable at every point within its domain. This smoothness implies that such functions have power series representations around every point in their domain.

An important characteristic of analytic functions is the Cauchy-Riemann equations, which link the real and imaginary parts of a complex function. If we let \(f(z) = u(x, y) + iv(x, y)\), where \(u\) and \(v\) are the real and imaginary parts respectively, then for \(f\) to be analytic, the following must hold:

\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.\]These equations ensure that the function is differentiable in the complex sense, making \(f(z)\) an analytic function.

vector fields

A vector field is a function that assigns a vector to each point in a subset of space. In this specific exercise, the vector field \(\mathbf{F}\) is given by \(\mathbf{F} = v \mathbf{i} + u \mathbf{j}\), where \(v\) and \(u\) are the scalar components defined by the analytic function \(f(z)\).

Vector fields are particularly useful for visualizing and analyzing multivariable phenomena. For example, they help in representing fluid flow or electromagnetic fields, showing both the magnitude and direction of the quantities at different points in the space.

An important task in working with vector fields is calculating their divergence and curl:

divergence and curl

The divergence and curl are two crucial operators for vector fields that measure different aspects of the field's behavior.

• Divergence: This measures the 'outflow' of a vector field from a point. In mathematical terms, the divergence of a vector field \(\mathbf{F} = v \mathbf{i} + u \mathbf{j}\) in two dimensions is given by:
  
  \[\operatorname{div} \mathbf{F} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}.\]
  For our vector field \(\mathbf{F}\) to be divergence-free (\( \operatorname{div} \mathbf{F} = 0 \)), this condition must hold:
  \[\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} = 0.\]
• Curl: This measures the rotational tendency of a vector field around a point. For the vector field \(\mathbf{F} \) in two dimensions, the curl is defined as:
  
  \[\operatorname{curl} \mathbf{F} = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}.\]
  For \(\mathbf{F}\) to be curl-free (\(\operatorname{curl} \mathbf{F} = 0\)), we need:
  \[\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} = 0.\]

The conditions where \(\operatorname{div} \mathbf{F} = 0\) and \(\operatorname{curl} \mathbf{F} = 0\) are equivalent to the Cauchy-Riemann equations. Thus, if an analytic function satisfies the Cauchy-Riemann equations, the associated vector field is both divergence-free and curl-free.