Problem 12 - Advanced Engineering Mathematics

Question

In Problems 7-14, express the given function in the form \(f(z)=u+i v .\) $$ f(z)=z^{4} $$

Step-by-Step Solution

Answer

\( f(z) = (x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3) \)

Step 1: Understanding the Problem

We need to express the function \( f(z) = z^4 \) in the form \( f(z) = u + iv \), where \( u \) and \( v \) are real-valued functions of \( x \) and \( y \). Here, \( z = x + iy \).

Step 2: Expanding \( z^4 \) using \( x + iy \)

Since \( z = x + iy \), we can write \( z^4 = (x + iy)^4 \). We need to expand this expression to separate the real and imaginary parts.

Step 3: Applying Binomial Theorem

Applying the binomial theorem to \( (x + iy)^4 \), we have: \[(x + iy)^4 = x^4 + 4x^3(iy) + 6x^2(iy)^2 + 4x(iy)^3 + (iy)^4 \]Next, compute each term.

Step 4: Simplifying Each Term

- \( x^4 \) is real.- \( 4x^3(iy) = 4ix^3y \) is imaginary.- \( (iy)^2 = -y^2 \) so \( 6x^2(iy)^2 = -6x^2y^2 \), which is real.- \( (iy)^3 = -iy^3 \) so \( 4x(iy)^3 = -4ixy^3 \) is imaginary.- \( (iy)^4 = y^4 \) is real.

Step 5: Combining Terms

Combine all the real and imaginary parts:\[f(z) = (x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3) \]Thus, \( u = x^4 - 6x^2y^2 + y^4 \) and \( v = 4x^3y - 4xy^3 \).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool that helps us expand expressions like \((a + b)^n\). This theorem states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This means we use the coefficients \(\binom{n}{k}\), also known as binomial coefficients, to distribute powers of \(a\) and \(b\).

In the context of our original problem, we apply the Binomial Theorem to expand \((x + iy)^4\). This gives us terms involving powers of \(x\) and \(iy\) with specific coefficients.

Each term can be computed individually:

• The first term is \(x^4\).
• The second term, \(4ix^3y\), comes from \(4x^3(iy)\).
• We have \(-6x^2y^2\) from \(6x^2(iy)^2\) where \((iy)^2 = -y^2\).
• Then \(-4ixy^3\) from \(4x(iy)^3\), where \((iy)^3 = -iy^3\).
• Finally, \(y^4\) from \((iy)^4 = y^4\), making it real.

By following this method, the complex expression neatly breaks down into real and imaginary parts.

Complex Variables

A complex variable, denoted as \(z\), usually takes the form \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

The world of complex variables allows us to extend the idea of traditional two-dimensional numbers to a two-part system, where one part is the real part and the other is the imaginary part.

This approach helps facilitate a wide variety of mathematical, physical, and engineering problems. Unlike real numbers, these can be visualized as vectors or points in a complex plane, where the horizontal axis represents the real part and the vertical axis the imaginary part.

• Real part: \(x\)
• Imaginary part: \(y\)

Thus, the complex variable becomes a key tool in more generalized solutions beyond the scope of real numbers alone. In our problem, understanding and using \(z = x + iy\) allows us to separate the function \(f(z) = z^4\) into distinct real and imaginary components.

Real and Imaginary Parts

Splitting functions into real and imaginary parts involves expressing a complex function \(f(z)\) as \(u + iv\), where both \(u\) and \(v\) are real-valued functions of \(x\) and \(y\).

In our specific problem, the objective is to break down \(f(z) = z^4\) into such parts:

• Start by expressing \(z\) as \(x + iy\).
• Use algebra and the Binomial Theorem to expand \((x + iy)^4\).

Upon expanding and simplifying, you will observe:

• The real part \(u = x^4 - 6x^2y^2 + y^4\)
• The imaginary part \(v = 4x^3y - 4xy^3\)

When boiled down to its essential real and imaginary components, each part plays a crucial role in how functions behave in complex analysis, affecting everything from graph visualization to solution approaches. This method is essential in tackling complex numbers in a very practical and comprehensible way.