Problem 4 - Advanced Engineering Mathematics

Question

In Problems 3-8, show that the given function is not analytic at any point. $$ f(z)=y+i x $$

Step-by-Step Solution

Answer

The function is not analytic because it does not satisfy the Cauchy-Riemann equations.

Step 1: Review the Conditions for Analyticity

For a function to be analytic at a point in the complex plane, it must satisfy the Cauchy-Riemann equations. This involves the partial derivatives of the real part and the imaginary part of the function.

Step 2: Identify the Real and Imaginary Parts

Given the function \( f(z) = y + ix \), identify the real and imaginary parts. Here, the real part \( u(x, y) = y \) and the imaginary part \( v(x, y) = x \).

Step 3: Compute Partial Derivatives

Calculate the partial derivatives: \( \frac{\partial u}{\partial x} = 0 \), \( \frac{\partial u}{\partial y} = 1 \), \( \frac{\partial v}{\partial x} = 1 \), and \( \frac{\partial v}{\partial y} = 0 \).

Step 4: Apply the Cauchy-Riemann Equations

According to the Cauchy-Riemann equations, \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Substituting the derivatives, we find: \( 0 eq 0 \) and \( 1 eq -1 \).

Step 5: Conclude Non-Analyticity

Since the Cauchy-Riemann equations are not satisfied at any point, the given function \( f(z) = y + ix \) is not analytic at any point in the complex plane.

Key Concepts

Cauchy-Riemann Equations

Understanding the Cauchy-Riemann equations is fundamental in complex analysis because they are used to determine if a function is analytic. A complex-valued function \( f(z) = u(x, y) + iv(x, y) \) is considered analytic at a point if it satisfies these specific conditions. These conditions, expressed as partial derivatives, are:

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

These equations ensure that the function exhibits certain smoothness and regularity near the point of interest and are usually tested for all points in a given region to conclude analyticity. If the equations do not hold at any point, the function is deemed non-analytic at that point. The precision of these derivatives plays a crucial role because even a minor discrepancy leads to non-analyticity.

Analytic Functions

An analytic function is one that is smooth and differentiable at every point in a given region of the complex plane. The differentiability here extends from real calculus but requires additional conditions due to the complex nature of the function. This extra layer of conditions is captured by the Cauchy-Riemann equations.In simpler terms, for \( f(z) \) to be analytic not only does it need to be differentiable in the traditional sense, but this differentiability must be consistent over all complex directions. This means if you take the derivative of the function along any path in the complex plane, the result remains the same.A function failing to meet these conditions at even a single point is classified as non-analytic at that point. This highlights why checking the Cauchy-Riemann equations is a fundamental step when studying complex functions.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus and apply to functions with more than one variable. For complex functions like \( f(z) = y + ix \), where \( z = x + iy \), the concept of partial derivatives helps separate the function into real and imaginary components: \( u(x, y) = y \) and \( v(x, y) = x \).Finding the partial derivative involves holding one variable constant while differentiating with respect to the other. For the function given, we calculate:

• \( \frac{\partial u}{\partial x} = 0 \) because \( u(x, y) = y \) only depends on \( y \).
• \( \frac{\partial u}{\partial y} = 1 \) because changing \( y \) changes \( u \) linearly.
• \( \frac{\partial v}{\partial x} = 1 \) where \( v(x, y) = x \) depends linearly on \( x \).
• \( \frac{\partial v}{\partial y} = 0 \) as \( v \) doesn’t change with \( y \).

Accurate computation of these derivatives is vital for applying the Cauchy-Riemann equations effectively and helps in understanding the function's behavior, especially regarding its analyticity.