Problem 6 - Advanced Engineering Mathematics

Question

Show that the given function is not analytic at any point. $$ f(z)=\bar{z}^{2} $$

Step-by-Step Solution

Answer

The function \( f(z) = \bar{z}^2 \) is not analytic at any point because the Cauchy-Riemann equations are not satisfied anywhere.

Step 1: Definition of Analytic Functions

A function is said to be analytic at a point if it possesses a derivative at that point, and there exists a neighborhood around that point where the function is differentiable. We will check if \( f(z) = \bar{z}^2 \) meets this condition.

Step 2: Express \( \bar{z} \) in terms of \( z \) and \( \overline{z} \)

The complex conjugate \( \bar{z} \) of a complex number \( z = x + yi \) is \( \bar{z} = x - yi \). Therefore, \( \bar{z}^2 = (x - yi)^2 \).

Step 3: Calculate \( \bar{z}^2 \) in terms of real and imaginary parts

Expand the expression: \( (x - yi)^2 = x^2 - 2xyi - (yi)^2 = x^2 - 2xyi - y^2i^2 = x^2 - 2xyi + y^2 \). This can be written as \( x^2 + y^2 - 2xyi \).

Step 4: Use the Cauchy-Riemann Equations

The Cauchy-Riemann equations for a function \( u(x, y) + iv(x, y) \) are \( u_x = v_y \) and \( u_y = -v_x \). For \( f(x, y) = (x^2 + y^2) - 2xyi \), \( u(x, y) = x^2 + y^2 \) and \( v(x, y) = -2xy \).

Step 5: Calculate Partial Derivatives

Compute the partial derivatives: \( u_x = 2x \), \( u_y = 2y \), \( v_x = -2y \), and \( v_y = -2x \).

Step 6: Check Cauchy-Riemann Equations

Substitute the derivatives into the Cauchy-Riemann equations: \( u_x = 2x eq -2x = -v_y \) and \( u_y = 2y eq 2y = -v_x \). These inequalities indicate that the Cauchy-Riemann equations are not satisfied at any point.

Step 7: Conclusion

Since the Cauchy-Riemann equations are not satisfied at any point, \( f(z) = \bar{z}^2 \) is not analytic at any point.

Key Concepts

Analyticity in Complex Analysis

In complex analysis, a function is considered analytic if it has a derivative at every point within a certain region or around a particular point. Essentially, it means that the function is smooth and well-behaved throughout that region. Analytic functions are incredibly important in complex analysis due to their nice properties. For instance, they can be represented as power series, which makes them predictable in their behavior. In this specific exercise, we are determining the analyticity of the function \( f(z) = \bar{z}^2 \). A function is analytic at a point if it is differentiable at every point in a neighborhood surrounding it. In our case, because \( f(z) = \bar{z}^2 \) does not satisfy this condition at any point, it is concluded that the function is not analytic anywhere.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two important equations used as a criterion to check whether a function is differentiable in the complex plane. For a function \( f(z) = u(x, y) + iv(x, y) \), where \( u \) and \( v \) are real-valued functions of two real variables \( x \) and \( y \), these equations are:

• \( u_x = v_y \)
• \( u_y = -v_x \)

These equations need to be satisfied for a function to be differentiable at that point in the complex plane. In our function \( f(z) = \bar{z}^2 \), given \( u(x, y) = x^2 + y^2 \) and \( v(x, y) = -2xy \), checking these equations shows that they are not satisfied at any point, confirming that \( f(z) \) is not analytic.

Complex Conjugates

Complex conjugation is an operation on complex numbers that changes the sign of the imaginary part. For a complex number \( z = x + yi \), its complex conjugate is \( \bar{z} = x - yi \). Complex conjugates are useful in a variety of calculations in complex analysis, especially in simplifying expressions and solving equations.

For the function \( f(z) = \bar{z}^2 \), the conjugation operation makes the function rely on the properties of the real parts being combined with their conjugates. By calculating, \( \bar{z}^2 = (x - yi)^2 \) becomes \( \bar{z}^2 = x^2 - 2xyi + y^2 \). This representation plays a key role in understanding why \( f(z) \) behaves differently than typical analytic functions, by affecting its ability to meet the criteria of the Cauchy-Riemann equations.

Differentiability in the Complex Plane

Differentiability in complex analysis takes on a slightly different meaning than in real analysis. In the complex plane, a function \( f(z) \) is differentiable at a point \( z_0 \) if the derivative \( f'(z_0) \) exists and is the same regardless of the path taken in approaching \( z_0 \). This path independence is the hallmark of complex differentiability and is a stronger condition than real differentiability.

For \( f(z) = \bar{z}^2 \), it is not differentiable at any point. This is because it fails to satisfy the necessary Cauchy-Riemann equations, which is a prerequisite for complex differentiability. The lack of differentiability throughout the complex plane means \( f(z) \) cannot have derivatives in the way required for analytic functions, making it non-analytic entirely.