Problem 8 - Advanced Engineering Mathematics

Question

\(u=\frac{x}{x^{2}+y^{2}}, v=\frac{y}{x^{2}+y^{2}} ; \quad \frac{\partial u}{\partial x}=\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}, \quad \frac{\partial v}{\partial y}=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} ; \quad \frac{\partial u}{\partial y}=-\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}=\frac{\partial v}{\partial x}\) The Cauchy-Riemann equations hold only at \((0,0) .\) since there is no neighborhood about \(z=0\) within which \(f\) is differentiable, we conclude \(f\) is nowhere analytic.

Step-by-Step Solution

Answer

The function \( f \) is not analytic anywhere.

Step 1: Understanding the Problem

We are given two functions, \( u = \frac{x}{x^{2}+y^{2}} \) and \( v = \frac{y}{x^{2}+y^{2}} \), and their partial derivatives. We need to verify the Cauchy-Riemann equations and determine where the function \( f = u + iv \) is analytic.

Step 2: Review Cauchy-Riemann Equations

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

Step 3: Check \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)

We have \( \frac{\partial u}{\partial x} = \frac{y^{2}-x^{2}}{(x^{2}+y^{2})^{2}} \) and \( \frac{\partial v}{\partial y} = \frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}} \). At any point where both functions are well-defined, these derivatives clearly are not equal, except when both \( x^2 \) and \( y^2 \) are zero, specifically at \((0,0)\).

Step 4: Check \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

We have \( \frac{\partial u}{\partial y} = -\frac{2xy}{(x^{2}+y^{2})^{2}} \) and \( \frac{\partial v}{\partial x} = \frac{2xy}{(x^{2}+y^{2})^{2}} \). These indeed satisfy the second Cauchy-Riemann condition as \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \) for all \( x \) and \( y \).

Step 5: Analyze the Entire Plane

The Cauchy-Riemann equations hold only at \((0,0)\) and there is no neighborhood around \(z=0\) where \(f\) satisfies both equations. The failure of the first Cauchy-Riemann condition elsewhere indicates \( f \) is not analytic anywhere else. Hence, \( f \) is nowhere analytic except potentially at \((0,0)\).

Key Concepts

Complex Analysis

Complex analysis is a beautiful branch of mathematics that studies functions of complex numbers. It focuses on complex functions, such as the form \( f = u + iv \), where \( u \) and \( v \) are real-valued functions of the real variables \( x \) and \( y \). In the simplest sense, a complex number is denoted as \( z = x + iy \), where \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).
This field is particularly interesting because it reveals the intricate relationship between real and imaginary components of a function. Complex analysis explores how these interconnected dimensions behave under various transformations, evaluating continuity, limits, and differentiability.
A key topic within complex analysis is the concept of differentiation, similar to differentiation in real analysis but with additional complexities due to the nature of complex numbers. Understanding complex analysis has immense applications ranging from engineering fields to theoretical physics.

Partial Derivatives

Partial derivatives are critical for understanding functions of several variables, like those encountered in complex analysis. Consider a function \( f(x, y) \) that depends on two variables \( x \) and \( y \). The partial derivative of \( f \) with respect to \( x \), denoted \( \frac{\partial f}{\partial x} \), measures how \( f \) changes as \( x \) is varied, while keeping \( y \) constant. Similarly, \( \frac{\partial f}{\partial y} \) measures changes as \( y \) varies, with \( x \) held constant.
If you think of a surface in three-dimensional space, calculating a partial derivative with respect to a variable is like slicing through that space parallel to one of the axes and observing how the surface shifts along that path.
Partial derivatives enable us to check if the Cauchy-Riemann equations hold, which are essential conditions for a complex function to be classified as analytic. They allow us to mathematically dissect how changes in one direction affect a multi-variable function, shedding light on complex behavior.

Analytic Functions

Analytic functions are central in complex analysis and hold much significance. A function, \( f(z) = u(x,y) + iv(x,y) \), is deemed analytic at a point if it is differentiable in some neighborhood around that point. More precisely, the partial derivatives of \( u \) and \( v \) must satisfy the Cauchy-Riemann equations:

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

In our problem, these conditions are only satisfied at the origin \((0, 0)\), demonstrating that the function \( f \) is not analytic elsewhere. An analytic function enjoys numerous powerful properties, such as possessing derivatives of all orders and being fully described by its Taylor series in its domain of analyticity.
The profound implication is that analytic functions often lead to neat solutions and simplifications, thanks to their smooth and predictable behavior. This makes them vital tools across various applications, from solving partial differential equations to signal processing and beyond.