Problem 12 - Advanced Engineering Mathematics

Question

Express the given function in the form \(f(z)=u+i v\) $$ f(z)=z^{4} $$

Step-by-Step Solution

Answer

The function can be expressed as: \(f(z) = (x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3)\).

Step 1: Express the Complex Number

Let the complex number be represented as \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part.

Step 2: Compute the Function

We need to find \(f(z) = (x + iy)^4\). Begin by expanding \((x + iy)^4\) using the binomial expansion theorem.

Step 3: Apply Binomial Theorem

The binomial expansion of \((x + iy)^4\) is given by: \[ (x + iy)^4 = x^4 + 4x^3(iy) + 6x^2(iy)^2 + 4x(iy)^3 + (iy)^4 \]

Step 4: Simplify Terms

Calculate each term in the expansion:- \(x^4\) remains as is.- \(4x^3(iy) = 4ix^3y\).- \(6x^2(iy)^2 = -6x^2y^2\) because \((i^2 = -1)\).- \(4x(iy)^3 = -4ixy^3\) because \((i^3 = -i)\).- \((iy)^4 = y^4\) because \((i^4 = 1)\).

Step 5: Combine Real and Imaginary Parts

Combine all terms to separate real and imaginary parts:\[(x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3)\]Here, \(u = x^4 - 6x^2y^2 + y^4\) and \(v = 4x^3y - 4xy^3\).

Step 6: Express Function in Required Form

Write the function in the form \(f(z) = u + iv\):\[f(z) = (x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3)\]

Key Concepts

Binomial Expansion

The binomial expansion is a crucial mathematical method used to expand expressions raised to a power. It's especially significant when dealing with powers involving complex terms. In our exercise, the function \[ f(z) = (x + iy)^4 \]is expanded using the binomial theorem.The binomial theorem provides a formula to expand expressions of the form \((a + b)^n\). For \((x + iy)^4\),we apply this theorem to break down the expression into a series of terms:

• \(x^4\)
• \(4x^3(iy)\)
• \(6x^2(iy)^2\)
• \(4x(iy)^3\)
• \((iy)^4\)

Each term is computed by using the binomial coefficients, which can be determined from Pascal's triangle or directly computed. The binomial expansion helps us separate and simplify complex expressions, turning the problem into manageable components.

Imaginary and Real Parts

To fully understand complex functions, it's essential to distinguish between their real and imaginary parts. Complex numbers are represented in the form \(z = x + iy\),where \(x\) is the real part and \(y\) is the imaginary part.

In our example of expanding \((x + iy)^4\),we identified and simplified each term using imaginary unit \(i\),where \(i^2 = -1\).Then, we categorized these terms into real and imaginary components:

• Real part, \(u\): \(x^4 - 6x^2y^2 + y^4\)
• Imaginary part, \(v\): \(4x^3y - 4xy^3\)

By separating terms with imaginary units from those without, we achieve the form \(f(z) = u + iv\).This distinct separation is vital for further complex analysis.

Complex Numbers

Complex numbers are fundamental in advanced mathematics, often used to solve real-world problems involving waves, oscillations, and other phenomena where two dimensions are needed.

A complex number, expressed as \(z = x + iy\),combines a real part \(x\) and an imaginary part \(y\). Imaginary numbers, characterized by the symbol \(i\) where \(i = \sqrt{-1}\),allow numbers to exist beyond the conventional real-number line.Key aspects:

• The real part \(x\) can be seen on a horizontal axis, while the imaginary part \(y\) appears on a vertical axis in the complex plane.
• Complex numbers can be represented, assessed, and manipulated using algebraic operations like addition and multiplication.
• When raising complex numbers to a power, as done in our exercise, the results require careful separation into real and imaginary components.

Understanding complex numbers and their properties is crucial for delving into higher mathematics and engineering fields.