Problem 34 - Advanced Engineering Mathematics

Question

Express \(\operatorname{Ln} z\) in the form \(a+i b\). \(z=(1+i)^{4}\)

Step-by-Step Solution

Answer

\(\ln z = \ln 4 + i\pi\).

Step 1: Find the modulus of the complex number

First, we find the modulus of the base complex number \((1 + i)\). The modulus is calculated as \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\).

Step 2: Express the complex number in polar form

Convert the complex number \(1+i\) to its polar form. The modulus is \(\sqrt{2}\) (from Step 1), and the argument \(\theta\) is determined by \(\tan^{-1}(\frac{1}{1}) = \frac{\pi}{4}\). Hence, in polar form, \(1+i = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\).

Step 3: Use De Moivre's Theorem

Apply De Moivre's theorem to \((1+i)^4\). We have: \((\sqrt{2})^4 (\cos(4 \cdot \frac{\pi}{4}) + i \sin(4 \cdot \frac{\pi}{4})) = 4(\cos \pi + i \sin \pi)\), which simplifies to \(-4\).

Step 4: Convert back to rectangular form

Recognize that \(-4\) in rectangular form is \(-4+0i\).

Step 5: Apply the logarithmic property

Now apply the complex logarithm formula: \(\ln z = \ln |z| + i\arg z\). For \(-4\), the modulus \(|z| = 4\) and \(\arg z = \pi\).

Step 6: Calculate the logarithm

Compute \(\ln 4 + i\pi\). So, \(\ln(-4) = \ln 4 + i\pi\).

Key Concepts

Complex Modulus

The complex modulus is a core concept when working with complex numbers. It is similar to finding the length of a vector in two dimensions.
The modulus of a complex number is indicated as |z|, where the complex number is in the form of a + bi. In mathematical terms, it is given by the formula: \[ |z| = \sqrt{a^2 + b^2} \]This formula calculates the distance of the complex number from the origin in the complex plane.

• For example, with z = 1 + i, the modulus is |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}.
• The modulus is always a non-negative real number.

Understanding and computing the modulus is essential for converting complex numbers into other forms, such as polar form.

Polar Form

The polar form of a complex number provides an alternative way of expressing complex numbers, especially useful for multiplication and division. Instead of using the rectangular (a + bi) form, the polar form utilizes the modulus and the argument (angle) of the number.

A complex number in polar form is expressed as: \[ z = r(\cos \theta + i \sin \theta) \]where:

• r is the modulus or magnitude, equivalent to |z|.
• \theta is the argument or angle.

For the complex number 1 + i,

• r = \sqrt{2}, and \theta = \frac{\pi}{4}, hence 1 + i can be written as \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}).

The polar form makes it easier to utilize De Moivre's Theorem in computations involving powers and roots of complex numbers.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for finding powers and roots of complex numbers when they are in polar form. This theorem states that for a complex number z in polar form:\[ (r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i \sin(n\theta)) \]This simplifies the computation of powers of complex numbers significantly.

For example, for (1+i)^4, you first express 1+i in polar form: \(\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\). Applying De Moivre's Theorem:\[ (\sqrt{2})^4 (\cos(4 \cdot \frac{\pi}{4}) + i \sin(4 \cdot \frac{\pi}{4})) = 4(\cos \pi + i \sin \pi) \] This simplifies to -4 (as \cos \pi = -1 and \sin \pi = 0). De Moivre's Theorem turns complex multiplication into manageable calculations.

Complex Numbers

Complex numbers extend the concept of the usual real numbers by introducing an imaginary unit denoted as i, where i^2 = -1. A complex number is typically represented in the form a + bi, where a is the real part and b is the imaginary part.

They are represented in a plane called the complex plane.

• The horizontal axis represents the real part, and the vertical axis represents the imaginary part.
• A point in this plane corresponds to a complex number.

Complex numbers are essential in many fields of science and engineering, allowing for solutions to equations that don't have real solutions. They can be manipulated and used in calculations involving powers and roots, with tools like De Moivre's Theorem facilitating these operations.