Problem 6 - Advanced Engineering Mathematics

Question

In Problems 3-8, 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 as it does not satisfy the Cauchy-Riemann equations.

Step 1: Understanding the Function

The given function is \( f(z) = \bar{z}^2 \), where \( \bar{z} \) is the complex conjugate of \( z \). In terms of \( z = x + iy \), the complex conjugate is \( \bar{z} = x - iy \). This means \( f(z) = (x - iy)^2 \).

Step 2: Calculating Derivatives

To check analyticity, we need to calculate the derivatives of the function. Analyticity in complex analysis means that the function must satisfy the Cauchy-Riemann equations. The partial derivatives must be calculated to verify this.

Step 3: Express \( f(z) \) in Terms of \( x \) and \( y \)

Expanding \( (x - iy)^2 \) gives \( x^2 - 2ixy - y^2 \). Therefore, \( f(z) = x^2 - y^2 - 2ixy \). Rewriting, we have \( u(x, y) = x^2 - y^2 \) and \( v(x, y) = -2xy \), where \( u \) and \( v \) are the real and imaginary parts respectively.

Step 4: Compute Partial Derivatives

Calculate the partial derivatives: - \( \frac{\partial u}{\partial x} = 2x \) - \( \frac{\partial u}{\partial y} = -2y \) - \( \frac{\partial v}{\partial x} = -2y \) - \( \frac{\partial v}{\partial y} = -2x \).

Step 5: Apply the Cauchy-Riemann Equations

The Cauchy-Riemann equations state that for analyticity, \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Here, we compare:- \( 2x eq -2x \)- \( -2y eq 2y \).Since these equations are not satisfied, the function is not analytic.

Key Concepts

Cauchy-Riemann Equations

In complex analysis, the Cauchy-Riemann equations are a pair of essential differential equations, a cornerstone of the subject. They tell us whether a complex function is analytic. For a function to be analytic (differentiable at every point in an open region), it must satisfy these equations:
\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]
where \( u(x, y) \) and \( v(x, y) \) are the real and imaginary parts of the complex function \( f(z) = u(x, y) + iv(x, y) \).

In the original exercise, the function \( f(z) = \bar{z}^2 \) is analyzed to check if it's analytic at any point by using these equations. Derivatives are computed, and it is shown that the equations are not satisfied. Therefore, \( f(z) = \bar{z}^2 \) is not analytic at any point.

Analytic Functions

An analytic function, often referred to as a holomorphic function, is a complex function that is differentiable at each point in its domain. This means, for the function \( f(z) \), a complex derivative exists at every point where it is analytic. Analytic functions are the main objects of study in complex analysis.

Key characteristics of analytic functions include:

• Satisfy the Cauchy-Riemann equations.
• They have a power series representation within the domain of analyticity.

These functions exhibit nice properties such as infinite differentiability and some interesting symmetries.

Since the original exercise demonstrates that the function \( f(z) = \bar{z}^2 \) doesn’t satisfy the Cauchy-Riemann equations, it cannot be analytic anywhere. Instead, it highlights the fact that not all generated functions from a complex number \( z \) and its conjugate are analytic, reminding us of the precise conditions needed for analyticity.

Complex Conjugate

The concept of a complex conjugate is foundational in understanding properties of complex numbers. The complex conjugate of a complex number \( z = x + iy \) is given as \( \bar{z} = x - iy \). It essentially "flips" the imaginary part to its negative, allowing for many applications in simplifying calculations, especially involving modulus and divisions.

Key points about complex conjugates:

• They maintain the real part of a complex number.
• The product \( z \cdot \bar{z} = x^2 + y^2 \) includes the squared modulus.
• Multiplication with the complex conjugate negates the imaginary component in expressions.

In the exercise, \( f(z) = \bar{z}^2 \), the use of complex conjugate shifts the notion of this function away from typical analytic properties, as showcased in functions composed of complex numbers. It exemplifies how transformations using conjugates can alter fundamental properties like differentiability.