Question

The function \(|\bar{z}|^{2}\) is not analytic at any point.

Short answer

The function \(|\bar{z}|^{2}\) is not analytic at any point because it doesn't satisfy the Cauchy-Riemann equations.

Step-by-step solution

Understand the term analytic function

A function is said to be analytic at a point if it is differentiable at that point and also in an open neighborhood around that point. In the context of complex functions, differentiability is defined using the limit, similar to real-valued functions.

Express the function

We are given the function \(|\bar{z}|^{2}\). In complex analysis, \(z = x + iy\) and \(\bar{z} = x - iy\), where \(x\) and \(y\) are the real and imaginary parts, respectively. Thus, \(|\bar{z}| = \sqrt{(x)^2 + (y)^2}\) and \(|\bar{z}|^{2} = (x^2 + y^2)\).

Check Cauchy-Riemann equations

For a function of a complex variable like \(f(z) = u(x,y) + iv(x,y)\) to be analytic, it must satisfy the Cauchy-Riemann equations \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\).

Identify real and imaginary components of the function

The function \(|\bar{z}|^{2}\) has no imaginary component (\(v(x, y) = 0\)) and its real component is \(u(x, y) = x^2 + y^2\).

Calculate partial derivatives

Calculate the partial derivatives: \(\frac{\partial u}{\partial x} = 2x\) and \(\frac{\partial u}{\partial y} = 2y\). Since \(v(x, y) = 0\), \(\frac{\partial v}{\partial x} = 0\) and \(\frac{\partial v}{\partial y} = 0\).

Check Cauchy-Riemann conditions

The Cauchy-Riemann equations require \(2x = 0\) and \(2y = 0\) for any point \((x,y)\). These conditions cannot be satisfied simultaneously unless \((x, y) = (0, 0)\). However, for a function to be analytic at a point, these conditions must hold in an open neighborhood, not just at a single point.

Conclude based on Cauchy-Riemann analysis

Since the Cauchy-Riemann equations do not hold for any open neighborhood in the complex plane, the function \(|\bar{z}|^{2}\) is not analytic at any point.

Key concepts

Analytic Function

In complex analysis, an analytic function is essentially one that has a derivative everywhere within its domain. This is akin to how we think about functions in real calculus, but it goes a step further with the properties of differentiability.
An analytic function is also known as holomorphic, and these functions have a fascinating hallmark—they are infinitely differentiable and can be represented by a power series.
This means that, locally, they look like polynomials, which gives them many useful properties.

• The function must be differentiable not just at a point but in an entire neighborhood around that point.
• Analytic functions exhibit smooth and well-behaved characteristics, allowing for many powerful mathematical tools to be applied.

Understanding whether a function is analytic involves checking the compliance with certain conditions like the Cauchy-Riemann equations.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are crucial for determining if a function of a complex variable is analytic. They are a set of two partial differential equations that connect the real and imaginary parts of the function. Suppose we have a complex function expressed as \(f(z) = u(x, y) + iv(x, y)\), where \(u\) is the real part and \(v\) is the imaginary part.
The equations are:

• \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\)
• \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\)

If both equations are satisfied, the function might be analytic at that point, given it satisfies these in an open neighborhood too. However, failure to satisfy the Cauchy-Riemann equations ensures the function cannot be analytic.
In the case of \(|\bar{z}|^2\), these conditions do not hold, hence it's not analytic at any point.

Complex Differentiability

Complex differentiability is a key characteristic that defines whether a function is analytic. A function is said to be complex differentiable at a point if it is differentiable in the complex sense—this involves limits and complex numbers.
For a function \(f(z) = u(x,y) + iv(x,y)\) to be complex differentiable, it must return a specific value regardless of the direction from which you approach the point \(z\) in the complex plane.

• This is much stricter compared to real differentiability, where we only consider direction along the real line.
• If a function is complex differentiable once, it’s guaranteed to be infinitely differentiable, making it analytic.

Failing the Cauchy-Riemann equations at any point undermines this differentiability, reinforcing that \(|\bar{z}|^2\) is non-analytic.

Real and Imaginary Parts

The real and imaginary parts of a complex function play a vital role in its analysis. Given a complex number \(z = x + iy\), its conjugate \(\bar{z}\) is \(x - iy\). For any complex function \(f(z) = u(x, y) + iv(x, y)\), \(u\) is the real part and \(v\) is the imaginary part.
Let’s examine \(|\bar{z}|^2 = x^2 + y^2\):

• Here, \(u(x, y) = x^2 + y^2\), presenting itself as the real part of the function.
• The imaginary component \(v(x, y)\) is zero, meaning the function has no imaginary component.

Real and imaginary components are integral to solving the Cauchy-Riemann equations. Without a non-zero imaginary part, satisfying these equations becomes impossible across the open neighborhoods necessary for analyticity.
This breakdown explains why \(|\bar{z}|^2\) fails to be analytic.