Problem 34 - Advanced Engineering Mathematics

Question

In Problems \(29-34\), express \(\operatorname{Ln} z\) in the form \(a+i b\). $$ z=(1+i)^{4} $$

Step-by-Step Solution

Answer

\(\operatorname{Ln} z = 2\ln 2 + i\pi\).

Step 1: Express the Complex Number in Polar Form

First, express the complex number \(1+i\) in polar form. The magnitude \(r\) can be calculated as \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\). The argument \(\theta\) is \(\tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}\). Thus, \(1+i = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)\).

Step 2: Raise to the Power 4 Using De Moivre's Theorem

Apply De Moivre's Theorem to raise the expression to the power of 4:\[ z = (\sqrt{2})^4 \left(\cos\left(4 \times \frac{\pi}{4}\right) + i\sin\left(4 \times \frac{\pi}{4}\right)\right) \]Simplify it to obtain:\[ z = 4 (\cos \pi + i \sin \pi) \]Further simplification yields:\[ z = -4 \] since \(\cos \pi = -1\) and \(\sin \pi = 0\).

Step 3: Convert \( -4 \) to Polar Form

Convert \(-4\) to polar form. The magnitude \(r\) is \(4\) and the argument \(\theta\) is \(\pi\). Therefore, \(-4 = 4 \left( \cos \pi + i \sin \pi \right)\).

Step 4: Compute the Natural Logarithm

Calculate the natural logarithm of \(z\) using \(\operatorname{Ln} z = \ln r + i \theta\). Here, \(r = 4\) and \(\theta = \pi\), thus:\[ \operatorname{Ln}(-4) = \ln 4 + i \pi \] which gives:\[ \operatorname{Ln}(-4) = 2\ln 2 + i \pi \] by simplifying \(\ln 4\) as \(2\ln 2\).

Step 5: Express \( \operatorname{Ln} z \) in the Form \( a + ib \)

Finally, express \(\operatorname{Ln} z\) as:\[ \operatorname{Ln} z = 2\ln 2 + i \pi \]Here, \(a = 2\ln 2\) and \(b = \pi\).

Key Concepts

Polar Form

When dealing with complex numbers, polar form is a way of expressing these numbers in terms of magnitude and direction. Complex numbers are generally represented in the Cartesian form as \(a + ib\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)).
In polar form, however, a complex number \(z\) is represented as \(r(\cos \theta + i\sin \theta)\), where:

• \(r\) is the magnitude of the complex number, given by \(\sqrt{a^2 + b^2}\).
• \(\theta\) is the argument, which indicates the angle the line from the origin to the point \((a, b)\) makes with the positive x-axis. It's calculated using \(\theta = \tan^{-1}\left( \frac{b}{a} \right)\).

This form is extremely useful for simplifying complex arithmetic operations, such as multiplication and division. It is also essential when applying De Moivre's Theorem. Converting to polar form can make working with powers and roots of complex numbers much simpler.

De Moivre's Theorem

De Moivre's Theorem provides a simple way to raise complex numbers in polar form to a power. It states that for any complex number \(z = r(\cos \theta + i \sin \theta)\) and a positive integer \(n\), the \(n\)th power of \(z\) can be expressed as:\[ z^n = r^n (\cos(n\theta) + i\sin(n\theta))\]This simplifies the problem of raising a complex number to a power. Instead of having to expand \((a + ib)^n\) using binomial expansion, you can just adjust the magnitude \(r\) and the angle \(\theta\).

• Example: For \((1+i)^4\), the initial step is converting \(1+i\) to its polar form, resulting in \(\sqrt{2} (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\).
• Applying De Moivre's Theorem, we raise the magnitude to the 4th power, and multiply the angle by 4, resulting in \(-4\).
• This makes computations involving powers of complex numbers both quicker and more intuitive.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are written in the form \(a + ib\), where \(a\) is the real part and \(ib\) is the imaginary part.
The imaginary unit \(i\) is defined such that \(i^2 = -1\). This definition expands the real number system to include solutions to equations that cannot be solved with only real numbers.

• Example: The square root of \(-1\), which is not a real number, is defined as \(i\).
• Complex numbers are used in various fields such as engineering, physics, and applied mathematics because of their ability to model waveforms and oscillations.

Through the introduction of the complex number system, all polynomial equations (according to the Fundamental Theorem of Algebra) have solutions. Complex logarithms, like those explored in the exercise, extend logarithms to complex numbers. Using properties of logarithms and polar coordinates, expressions like \(\ln(a + ib)\) become manageable.