Problem 21 - Complex Analysis

Question

Where does the function \(f: \mathbb{C}^{*} \rightarrow \mathbb{C}, f(z)=z \bar{z}+z / \bar{z}\), satisfy the CAUCHYRIEMANN differential equations?

Step-by-Step Solution

Answer

The function satisfies the Cauchy-Riemann equations when \( y = 0 \).

Step 1: Understand the Function

The given function is \( f(z) = z\bar{z} + \frac{z}{\bar{z}} \). Here, \( z = x + iy \) where \( x \) and \( y \) are real numbers, and \( \bar{z} = x - iy \). We need to check where this function satisfies the Cauchy-Riemann equations.

Step 2: Express the Function in Terms of Real Components

Express the function \( f(z) = z\bar{z} + \frac{z}{\bar{z}} \) in terms of \( x \) and \( y \).\[ z\bar{z} = (x + iy)(x - iy) = x^2 + y^2 \]\[ \frac{z}{\bar{z}} = \frac{x+iy}{x-iy}. \]To divide complex numbers multiply numerator and denominator by the conjugate of the denominator.\[ \frac{(x + iy)(x + iy)}{(x - iy)(x + iy)} = \frac{x^2 + 2xyi - y^2}{x^2 + y^2} = \frac{x^2 - y^2}{x^2+y^2} + i\frac{2xy}{x^2 + y^2} \].So, \( f(z) = x^2 + y^2 + \frac{x^2 - y^2}{x^2+y^2} + i\frac{2xy}{x^2 + y^2} \).

Step 3: Separate into Real and Imaginary Parts and Identify Components

Separate the function \( f(z) = u(x,y) + iv(x,y) \):\[ u(x,y) = x^2 + y^2 + \frac{x^2 - y^2}{x^2 + y^2} \] \[ v(x,y) = \frac{2xy}{x^2 + y^2} \].

Step 4: Apply Cauchy-Riemann Equations

The Cauchy-Riemann equations are given by: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).Calculate \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \) and equate accordingly to see where they are satisfied.

Step 5: Calculate the Necessary Partial Derivatives

Calculate the partial derivatives.\[ \frac{\partial u}{\partial x} = 2x + \frac{2x(x^2 - y^2) - 2x^3 + 2xy^2}{(x^2 + y^2)^2} = 2x + \frac{-4x^3 + 4xy^2}{(x^2 + y^2)^2} \].\[ \frac{\partial u}{\partial y} = 2y - \frac{2y}{(x^2 + y^2)^2} \times (-x^2 + y^2) = 2y + \frac{4xy^2 - 4y^3}{(x^2 + y^2)^2}\].\[ \frac{\partial v}{\partial x} = \frac{2y(x^2 - y^2)}{(x^2 + y^2)^2} \].\[ \frac{\partial v}{\partial y} = \frac{2x(x^2 - y^2)}{(x^2 + y^2)^2} - \frac{4x^3y}{(x^2 + y^2)^2} \].

Step 6: Finding Where Derivatives Satisfy Cauchy-Riemann

Plug in the partial derivatives into the Cauchy-Riemann equations and simplify:\[ \frac{\partial u}{\partial x} = \frac{2x(x^2 - y^2) - 2x^3 + 2xy^2}{(x^2 + y^2)^2} \] should equal \( \frac{2x(x^2 - y^2)}{(x^2 + y^2)^2} \). Simplifying reveals this holds when \( y=0 \), so the second equation resolves trivially.

Step 7: Confirm All Conditions

Upon examination, the conditions of the Cauchy-Riemann equations hold true only when components from step 6 cancel, requiring trivial conditions such as \( y=0 \). As all needed cancellations rely on \( v \equiv 0 \) with no imaginary components, confirming that the region will be limited or narrow.

Key Concepts

Complex Functions

Complex functions are functions that take complex numbers as input and produce complex numbers as output. In our context, we have a function of the form \( f(z) = z\bar{z} + \frac{z}{\bar{z}} \). The variable \( z \) represents a complex number given by \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit. The conjugate of \( z \), denoted as \( \bar{z} \), is \( x - iy \).

This function involves operations like multiplication, division, and addition of complex numbers. Complex conjugation, as seen with \( \bar{z} \), flips the imaginary part to its negative. Our goal is to understand how this function behaves and satisfies specific criteria when examined through the lens of complex analysis.

• Understanding Complex Numbers: They are represented as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
• Basic Operations: Functions may involve arithmetic like addition, subtraction, multiplication, and division of these numbers.
• Complex Conjugate: Given \( z = x + iy \), its conjugate is \( \bar{z} = x - iy \).

By expressing functions like \( f(z) \) in terms of real and imaginary parts, we can analyze their behavior more closely.

Partial Derivatives

Partial derivatives involve taking derivatives of functions with respect to one variable while holding others constant. It is an essential tool in analyzing the behavior of multi-variable functions.

In our function \( f(z) = u(x,y) + iv(x,y) \), we separate it into real component \( u(x,y) \) and imaginary component \( v(x,y) \). The Cauchy-Riemann equations require calculating the partial derivatives of \( u \) and \( v \) with respect to \( x \) and \( y \).

• Deriving \( u \): Begin by finding \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \). This helps us look at how changes in \( x \) or \( y \) affect \( u(x,y) \).
• Deriving \( v \): Similarly, find \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \).

Apply these derivatives to the Cauchy-Riemann equations:

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

These equations need to hold true for the function to be differentiable in the complex sense. This step ensures that we understand the changes and dependencies between the real and imaginary parts of the function.

Complex Analysis

Complex analysis studies functions that take complex numbers as inputs. A key aspect of this field involves ensuring that functions satisfy conditions such as the Cauchy-Riemann equations to be differentiable.

The Cauchy-Riemann equations are pivotal in confirming the analyticity of complex functions. They express that the partial derivatives of a complex function must balance in a particular way:

• Criteria for Differentiability: A complex function is differentiable only if the Cauchy-Riemann equations are satisfied across a domain.
• Application in Our Example: The function \( f(z) \), upon applying these equations, demonstrates that it holds under the condition \( y = 0 \). It shows that these points potentially lie along certain lines or areas where the function remains differentiable.

Complex analysis allows mathematicians to delve deeper into the properties and behaviors of these functions, exploring beyond real analysis by examining angles, morphisms, and more. Understanding when and where these equations hold brings insight into both the limitations and capabilities of the function under study.