Problem 4 - Advanced Engineering Mathematics

Question

$$u=y, \quad v=x ; \quad \frac{\partial u}{\partial x}=0=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=1,-\frac{\partial v}{\partial x}=-1 . \text { since } 1 \neq-1, f \text { is not analytic at any point. }$$

Step-by-Step Solution

Answer

The function is not analytic because the Cauchy-Riemann conditions are not met.

Step 1: Understanding the Given Variables

We are provided with two variables: \( u = y \) and \( v = x \). These form the real (\( u \)) and imaginary (\( v \)) parts of a potential complex function \( f(z) = u + iv \).

Step 2: Calculating Partial Derivatives

Compute the partial derivatives: \( \frac{\partial u}{\partial x} = 0 \), \( \frac{\partial v}{\partial y} = 0 \), \( \frac{\partial u}{\partial y} = 1 \), and \( -\frac{\partial v}{\partial x} = -1 \). These derivatives will be crucial for checking the Cauchy-Riemann equations.

Step 3: Checking Cauchy-Riemann Equations

For \( f \) to be analytic, Cauchy-Riemann equations need to hold: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). We have \( 0 = 0 \) for the first equation. However, for the second, \( 1 eq -1 \), which means this condition does not hold.

Step 4: Conclusion: Analyticity

Since the Cauchy-Riemann equations are not satisfied, the function \( f(z) = u + iv \) is not analytic at any point.

Key Concepts

Cauchy-Riemann Equations

In complex analysis, the Cauchy-Riemann equations are fundamental because they determine whether a function is analytic. This means the function is differentiable at every point in its domain and has a complex derivative. 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 \( y \), the Cauchy-Riemann equations are:

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

These must both be satisfied for the function to be considered analytic. In simpler terms, these equations relate the partial derivatives of the real and imaginary parts of a complex function. If they hold at a point, the function has a derivative at that point, which is a key indicator of analyticity.

In the context of our exercise, both equations were checked but only the first equation held true. It was shown that \( \frac{\partial u}{\partial y} = 1 \) was not equal to \( -\frac{\partial v}{\partial x} = -1 \), violating the second Cauchy-Riemann equation. As a result, the function \( f \) is not analytic.

Analytic Functions

Analytic functions are a special class of functions in complex analysis that are not only differentiable but also possess derivatives of all orders. Their beauty lies in their smoothness and the fact that they can be expressed as power series. Notably, these functions are typically referred to as holomorphic, especially when defined in a domain.

For a function \( f(z) \) to be analytic at a point \( z_0 \), it must satisfy several criteria:

• Existence of a derivative at every point close to \( z_0 \)
• Satisfaction of the Cauchy-Riemann equations in a neighborhood around \( z_0 \)
• Continuous partial derivatives

Fulfillment of these conditions ensures that the function is smooth and behaves predictably near each point. However, if any of these conditions are violated, the function is deemed non-analytic or not part of the analytic family, as demonstrated by our exercise function.

Partial Derivatives

Partial derivatives are essential tools in mathematics, particularly in complex analysis. They help us understand how a function changes when only one of its input variables changes while keeping everything else constant. This is especially useful in multivariable functions like those seen in complex analysis where both real and imaginary parts are functions of two real variables \( x \) and \( y \).

In our exercise, the partial derivatives of the real part \( u \) with respect to \( x \), and \( y \), and the partial derivatives of the imaginary part \( v \) with respect to \( x \), and \( y \) were calculated. These derivatives enabled us to check the Cauchy-Riemann equations to determine the analytic nature of the function. Understanding partial derivatives is crucial because they give insight into the function's behavior and are fundamental in verifying whether the essential conditions for analyticity are satisfied.