Problem 38 - Advanced Engineering Mathematics

Question

Determine whether the function $$ f(z)=x^{3}+x y^{2}-4 x+i\left(4 y-y^{3}-x^{2} y\right) $$ is differentiable. Is it analytic?

Step-by-Step Solution

Answer

The function is differentiable on the circle \( x^2 + y^2 = 2 \) but not analytic.

Step 1: Understanding the Function

The given function is a complex function of the form \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) = x^3 + xy^2 - 4x \) and \( v(x, y) = 4y - y^3 - x^2y \). We must check the Cauchy-Riemann equations to determine differentiability.

Step 2: Checking the Cauchy-Riemann Equations

The Cauchy-Riemann equations require \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Calculate these partial derivatives: \( \frac{\partial u}{\partial x} = 3x^2 + y^2 - 4 \), \( \frac{\partial u}{\partial y} = 2xy \), \( \frac{\partial v}{\partial x} = -2xy \), and \( \frac{\partial v}{\partial y} = 4 - 3y^2 - x^2 \).

Step 3: Evaluating the Cauchy-Riemann Equations

To satisfy the Cauchy-Riemann equations, set \( 3x^2 + y^2 - 4 = 4 - 3y^2 - x^2 \) and \( 2xy = -(-2xy) \). Simplifying the first equation gives \( 4x^2 + 4y^2 = 8 \), leading to \( x^2 + y^2 = 2 \). The second equation is an identity, so it holds true generally.

Step 4: Conclusion on Differentiability and Analyticity

The Cauchy-Riemann equations are satisfied only along the circle defined by \( x^2 + y^2 = 2 \). This restriction means \( f(z) \) is differentiable only on this circle and therefore, not analytic everywhere in the complex plane.

Key Concepts

Cauchy-Riemann Equations

When analyzing complex functions for differentiability, the Cauchy-Riemann equations play a crucial role. These equations are conditions that must be met for a complex function to be differentiable at a point. The function is expressed as

• Real part: \( u(x, y) \)
• Imaginary part: \( v(x, y) \)

For differentiability, two partial derivative conditions need to be satisfied:

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

These conditions ensure the function's complex derivative exists at the point. Referencing the exercise, examine the partial derivatives:

• For \( u \), the partial derivatives are \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \)
• For \( v \), the partial derivatives are \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \)

Once the expressions for these derivatives are plugged into the Cauchy-Riemann conditions, it allows for determining whether these are satisfied on specific loci in the complex plane.

Differentiability

Differentiability of a complex function at a point indicates that the function has a well-defined and continuous derivative there. In the realm of complex analysis, this requires more than just the existence of derivatives like in real analysis. It signifies a harmony between the real and imaginary components governed by the Cauchy-Riemann equations.

In the exercise, the differentiation is examined by setting the equations from the Cauchy-Riemann conditions. The solution finds that the complex function \( f(z) \) can be differentiated at points on the locus \( x^2 + y^2 = 2 \). This means:

• The derivative exists at these points only, resulting in "partial" differentiability.
• Outside of this circle, the function cannot be differentiated in the complex sense.

This reflects how complex differentiability is stricter compared to real differentiation, requiring the function to simultaneously adhere to multiple differential criteria.

Analyticity

A function is considered analytic if it is differentiable at every point in an open set or throughout a domain in the complex plane. This goes beyond merely finding isolated points of differentiability. Analytic functions possess smooth curves and conform to the complex plane's rules. They can be represented by power series, which makes them very powerful in complex analysis.

However, in the provided exercise, \( f(z) \) exhibits differentiability only along a specific circular boundary \( x^2 + y^2 = 2 \). Due to the restriction to this geometric boundary, the function cannot be analytic everywhere.

• The underlying reason for the non-analytic nature is the failure of Cauchy-Riemann equations outside the circle.
• To be analytic, the function should satisfy differentiability conditions across entire open regions, not just pointwise or on isolated curves.

Thus, its analyticity is limited or non-existent when considering the requirements for a function to be analytic across a broader region in the complex plane.