Problem 19 - Advanced Engineering Mathematics

Question

\\[ u=x^{3}+3 x y^{2}-x, v=y^{3}+3 x^{2} y-y ; \quad \frac{\partial u}{\partial x}=3 x^{2}+3 y^{2}-1, \frac{\partial v}{\partial y}=3 y^{2}+3 x^{2}-1 ; \quad \frac{\partial u}{\partial y}=6 x y,-\frac{\partial v}{\partial x}=-6 x y \\] \(u\) and \(v\) are continuous and have continuous first partial derivatives. The Cauchy-Riemann equations are satisfied whenever \(6 x y=-6 x y\) or \(12 x y=0 .\) The points satisfying \(12 x y=0\) are the points along the \(y\) -axis \((x=0)\) or along the \(x\) -axis \((y=0) .\) The function \(f\) is differentiable but not analytic on either axis; there is no neighborhood about any point \(z=x\) or \(z=i y\) within which \(f\) is differentiable.

Step-by-Step Solution

Answer

The function \( f \) is differentiable but not analytic along the \(x\)-axis and \(y\)-axis.

Step 1: Identify the problem

We need to verify whether the given functions \( u \) and \( v \) satisfy the Cauchy-Riemann equations and determine where the function \( f \) is differentiable and analytic.

Step 2: Analyze the first-order partial derivatives

We will use the given partial derivatives: \( \frac{\partial u}{\partial x} = 3x^2 + 3y^2 - 1 \), \( \frac{\partial u}{\partial y} = 6xy \), \( \frac{\partial v}{\partial y} = 3y^2 + 3x^2 - 1 \), \( -\frac{\partial v}{\partial x} = -6xy \) to check compliance with Cauchy-Riemann equations.

Step 3: Apply the Cauchy-Riemann equations

The Cauchy-Riemann equations state that \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Given \( 6xy = -6xy \), these equations are satisfied only when \( 12xy = 0 \).

Step 4: Identify where Cauchy-Riemann equations hold

From \( 12xy = 0 \), we gather that \( x = 0 \) or \( y = 0 \). This implies the Cauchy-Riemann equations are satisfied along the \(x\)-axis and \(y\)-axis.

Step 5: Determine differentiability and analyticity

Since the Cauchy-Riemann equations hold on the \(x\)-axis and \(y\)-axis only, \( f \) is differentiable along these axes. However, because there is no neighborhood around any point where \( f \) satisfies Cauchy-Riemann entirely, it is not analytic on the axes.

Key Concepts

Complex Differentiability

The notion of complex differentiability is central in complex analysis. To be complex differentiable at a point, a complex function must satisfy a stricter condition compared to real differentiability. For a function \( f(z) = u(x, y) + iv(x, y) \), where \( u \) and \( v \) are real-valued functions, history tells us that it needs to satisfy the

• Cauchy-Riemann equations: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

These conditions ensure that the function behaves 'nicely' at that point and that the complex derivative exists.
In our exercise, these conditions were considered using the partial derivatives of \( u \) and \( v \). The equations are met when the value of \( x \) or \( y \) is zero, meaning that the function \( f \) is differentiable along the axes. However, for a function to be analytic, it should hold true not just at isolated points but within a neighborhood, which is not the case here.

Partial Derivatives

Partial derivatives play a crucial role in understanding how multi-variable functions change. When dealing with a function \( f(x, y) \), partial derivatives tell us how \( f \) changes as either \( x \) or \( y \) varies while keeping the other variable constant.

• First-order Partial Derivatives: These were calculated as \( \frac{\partial u}{\partial x} = 3x^2 + 3y^2 - 1 \), \( \frac{\partial u}{\partial y} = 6xy \), \( \frac{\partial v}{\partial y} = 3y^2 + 3x^2 - 1 \), \( -\frac{\partial v}{\partial x} = -6xy \).

By substituting these into the Cauchy-Riemann equations, we test whether the function \( f \) is complex differentiable. It tells us about the rate of change of \( u \) and \( v \) at a specific point. These derivatives were essential in verifying that the Cauchy-Riemann equations are met along the axes \( x = 0 \) and \( y = 0 \). This implies that \( f \) is differentiable at these points.

Analytic Functions

Analytic functions, often called holomorphic, are a class of functions in complex analysis known for their own special properties. A complex function is said to be analytic at a point if it is complex differentiable across an entire neighborhood of that point, not just at the point itself.

• Cauchy-Riemann in Neighborhoods: For our given function \( f \), even though it was differentiable at points along the axes, for it to be considered analytic, it needed to satisfy the Cauchy-Riemann equations in some neighborhood around those points.
• Insights from the Exercise: The function \( f \) fails to be analytic since those conditions aren't fully met beyond those immediate lines. Hence, it's only locally differentiable along isolated points on the \( x \)-axis and \( y \)-axis.

In essence, to be analytic, a function must be complex differentiable everywhere in any open set containing the point, presenting a richer and more stringent notion than regular differentiability. This exercise highlights the thin line between being differentiable at a point and being fully analytic.