Problem 1 - Advanced Engineering Mathematics

Question

In Problems 1 and 2 , the given function is analytic for all \(z\). Show that the Cauchy-Riemann equations are satisfied at every point. $$ f(z)=z^{3} $$

Step-by-Step Solution

Answer

The Cauchy-Riemann equations are satisfied for all points; therefore, \( f(z) = z^3 \) is analytic everywhere.

Step 1: Define the function in terms of x and y

Let the complex number \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit. The function \( f(z) = z^3 \) can be expressed as \( f(z) = (x + iy)^3 \).

Step 2: Expand the function

To find \( f(x, y) \), expand \( (x + iy)^3 \) using the binomial theorem: \[ (x + iy)^3 = x^3 + 3x^2(iy) + 3x(iy)^2 + (iy)^3. \] Simplifying further, we have: \[ =x^3 + 3x^2(iy) - 3xy^2 - iy^3. \] Collecting real and imaginary parts, \( f(z) = (x^3 - 3xy^2) + i(3x^2y - y^3) \). Let \( u(x, y) = x^3 - 3xy^2 \) and \( v(x, y) = 3x^2y - y^3 \).

Step 3: Verify Cauchy-Riemann equations

The Cauchy-Riemann equations are: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \] Compute the partial derivatives: \[ \frac{\partial u}{\partial x} = 3x^2 - 3y^2, \quad \frac{\partial v}{\partial y} = 3x^2 - 3y^2, \] \[ \frac{\partial u}{\partial y} = -6xy, \quad \frac{\partial v}{\partial x} = 6xy. \] Check if they satisfy the Cauchy-Riemann equations: \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Both equations hold true.

Step 4: Conclusion

Since the Cauchy-Riemann equations are satisfied for the function \( f(z) = z^3 \) at every point, the function is analytic, as expected, for all \( z \).

Key Concepts

Analytic Functions

An analytic function is a complex function that is smooth and differentiable at each point in its domain. This means it has a derivative everywhere within an open subset of the complex plane.
When a function is analytic, it must satisfy certain conditions, known as the Cauchy-Riemann equations, which are crucial in complex analysis.
Analytic functions are incredibly important because they behave nicely and have many interesting properties, such as being infinitely differentiable and having power series representations.

• To check if a function is analytic, we typically verify if the Cauchy-Riemann equations are met at every point.
• One key feature of analytic functions is that their behavior is locally determined by their derivatives.
• Analytic functions are also related to harmonic functions, which are solutions to Laplace's equation.

When working with analytic functions, always remember that these functions possess a level of smoothness that allows for various powerful theorems, such as Cauchy's integral theorem, to hold.

Complex Functions

Complex functions involve a variable or variables that take on complex values. These are crucial in complex analysis, a field that extends principles from real-number calculus to the complex numbers.
Complex numbers take the form of \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).
This format allows both real and imaginary components of the function to interact in interesting ways, leading to rich structures and results, such as Euler's formula and the concept of residues in complex integration.

• Complex functions can be visualized in the complex plane, where each complex number is a point.
• These functions are pivotal in various fields, including engineering, physics, and applied mathematics.
• Complex functions include essential operations like addition, multiplication, and complex conjugation, affecting both the real and imaginary parts.

Understanding complex functions and their properties is key to solving many advanced problems in mathematics and related disciplines.

Partial Derivatives

Partial derivatives are the derivatives of functions with more than one variable, where we take the derivative with respect to one variable while keeping others constant.
In the context of complex functions, you often see derivatives with respect to \( x \) or \( y \), the real and imaginary parts, as in the Cauchy-Riemann equations.
These equations require the partial derivatives to be equal in a certain way, ensuring the complex function is analytic.

• For a function \( u(x, y) \), \( \frac{\partial u}{\partial x} \) denotes the rate of change of \( u \) with respect to \( x \) while keeping \( y \) constant.
• Similarly, \( \frac{\partial u}{\partial y} \) is how \( u \) changes with \( y \), holding \( x \) constant.
• Practically, these derivatives help analyze the behavior of functions and are used in optimizing functions and solving differential equations.

By mastering partial derivatives, you unlock the ability to explore more complex problems involving multi-variable functions and gain a deeper understanding of their graphical behavior and sensitivity to changes in one direction.