Problem 21 - Advanced Engineering Mathematics

Question

$$u=e^{x} \cos y, \quad v=-e^{x} \sin y ; \quad \frac{\partial u}{\partial x}=e^{x} \cos y, \quad \frac{\partial v}{\partial y}=-e^{x} \cos y ; \quad \frac{\partial u}{\partial y}=-e^{x} \sin y, \quad-\frac{\partial v}{\partial x}=e^{x} \sin y$$ Since the Cauchy-Riemann equations are not satisfied at any point, \(f\) is nowhere analytic.

Step-by-Step Solution

Answer

The function \(f\) is nowhere analytic because the Cauchy-Riemann equations are not satisfied.

Step 1: Understand the Context

We are given two functions, \(u\) and \(v\), and asked to check if \(f(z) = u + iv\) is analytic. This requires using the Cauchy-Riemann equations to determine if they hold for the given functions.

Step 2: Recall the Cauchy-Riemann Equations

The Cauchy-Riemann equations state that for a function \(f = u + iv\) to be analytic, \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\) must be true.

Step 3: Compare Given Partial Derivatives

From the problem, we have \(\frac{\partial u}{\partial x} = e^{x} \cos y\) and \(\frac{\partial v}{\partial y} = -e^{x} \cos y\). We also have \(\frac{\partial u}{\partial y} = -e^{x} \sin y\) and \(-\frac{\partial v}{\partial x} = e^{x} \sin y\).

Step 4: Verify First Cauchy-Riemann Equation

Check if \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\). Accordingly, we compare \(e^{x} \cos y\) and \(-e^{x} \cos y\). They are not equal, hence the equation is not satisfied.

Step 5: Verify Second Cauchy-Riemann Equation

Check if \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\). We compare \(-e^{x} \sin y\) and \(e^{x} \sin y\). They are not equal, so this equation is not satisfied either.

Step 6: Conclude Analyticity

Since neither of the Cauchy-Riemann equations is satisfied anywhere, the function \(f\) is not analytic at any point.

Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are fundamental in determining the analyticity of complex functions. They present conditions that must be fulfilled for a function defined by a complex variable. In the context of complex analysis, if a function \( f(z) = u + iv \) is to be considered analytic, the following conditions must hold throughout a domain:

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

These equations connect the real part \( u \) and the imaginary part \( v \) of the complex function with their partial derivatives.
If both conditions are met in a region, the function is analytic in that region. If they fail at any point, the function is not analytic there.

Analytic Functions

Analytic functions are central in complex analysis because they possess some remarkable properties. These functions are differentiable in a complex sense across a region or domain. Being analytic implies that a function is smooth and has derivatives of all orders, making them predictable and well-behaved.
For a function \( f(z) = u + iv \) to be analytic, the Cauchy-Riemann equations must be satisfied. This requirement ties the concept of complex differentiability to the partial derivatives of \( u \) and \( v \).

• Smoothness: Analytic functions are infinitely differentiable.
• Local Behavior: The behavior of an analytic function locally mirrors that of a real-valued polynomial.
• Convergence: They can often be expressed as convergent power series.

Ensuring these qualities means that analytic functions play a vital role in many applications within mathematics and engineering.

Partial Derivatives

Partial derivatives are a form of differentiation when dealing with multivariable functions. They measure how a function changes as one variable varies while others are held constant. In the context of complex functions, partial derivatives are crucial for checking the criteria for analyticity, especially through the Cauchy-Riemann equations.
For a function \( u(x, y) \), the partial derivative with respect to \( x \), denoted \( \frac{\partial u}{\partial x} \), and with respect to \( y \), denoted \( \frac{\partial u}{\partial y} \), give rates of change in the directions of \( x \) and \( y \), respectively.
They allow you to understand how the components \( u \) and \( v \) independently behave. Key to this analysis is:

• Identifying gradient-like properties, representing directional change.
• Influencing conditions for complex differentiability.

In essence, mastering partial derivatives provides insight into how multivariable functions operate, especially in complex analysis.