Problem 26 - Advanced Engineering Mathematics

Question

In Problems 23-28, express \(\ln z\) in the form \(a+i b\). $$ z=1+i $$

Step-by-Step Solution

Answer

\( \ln(1+i) = \frac{1}{2} \ln 2 + i \frac{\pi}{4} \).

Step 1: Identify the Formula for Complex Logarithms

For a complex number \( z = x + iy \), its logarithm can be expressed as \( \ln z = \ln |z| + i \arg(z) \). Here, \(|z|\) is the modulus and \( \arg(z) \) is the argument of the complex number.

Step 2: Compute the Modulus of the Complex Number

To find \( |z| \) for \( z = 1 + i \), use the formula: \( |z| = \sqrt{x^2 + y^2} \). So, \(|z| = \sqrt{1^2 + 1^2} = \sqrt{2}\).

Step 3: Determine the Argument of the Complex Number

The argument \( \arg(z) \) is given by \( \tan^{-1}(\frac{y}{x}) \) where \( x = 1 \) and \( y = 1 \). Thus, \( \arg(z) = \tan^{-1}(\frac{1}{1}) = \tan^{-1}(1) = \frac{\pi}{4} \) radians.

Step 4: Assemble the Logarithmic Expression

Substitute \( |z| \) and \( \arg(z) \) into the logarithm expression: \( \ln z = \ln |z| + i \arg(z) = \ln \sqrt{2} + i \frac{\pi}{4} \). By simplifying \( \ln \sqrt{2} \), we have \( \ln \sqrt{2} = \frac{1}{2} \ln 2 \).

Step 5: Finalize the Expression into \(a + ib\) form

Thus, \( \ln z = \frac{1}{2} \ln 2 + i \frac{\pi}{4} \). Here, \( a = \frac{1}{2} \ln 2 \) and \( b = \frac{\pi}{4} \).

Key Concepts

Understanding Complex Numbers

Complex numbers are a fundamental concept in mathematics that extends the idea of the one-dimensional number line to the complex plane. A complex number is generally expressed in the form \(z = x + iy\), where \(x\) is the real part, \(y\) is the imaginary part, and \(i\) is the imaginary unit with the remarkable property \(i^2 = -1\). These numbers are visualized in a two-dimensional plane called the complex plane, where:

• The horizontal axis represents the real part \(x\)
• The vertical axis represents the imaginary part \(y\)

This visualization helps in understanding complex number operations, as well as their magnitude and direction when translated into polar coordinates.
Complex numbers are vital for solving polynomial equations that do not have real roots, such as \(x^2 + 1 = 0\). They are widely used in various fields, including engineering, physics, computer science, and applied mathematics.

Modulus and Argument

In the realm of complex numbers, the terms **modulus** and **argument** are pivotal.The modulus, denoted as \(|z|\), describes the magnitude of a complex number. It is the distance from the origin to the point \((x, y)\) in the complex plane. This is calculated using:

• \(|z| = \sqrt{x^2 + y^2}\)

For our problem with \(z = 1 + i\), the modulus is \(\sqrt{2}\).
The argument, denoted as \(\arg(z)\), represents the angle the vector from the origin to the point \((x, y)\) makes with the positive real axis. It adjusts the direction of the complex number and is evaluated as:

• \(\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)\)

In this case, the argument is \(\frac{\pi}{4}\) radians, indicating the directional rotation of the complex number.
Understanding modulus and argument is crucial for expressing complex numbers in polar form, which simplifies multiplication and division operations.

Euler's Formula and Complex Numbers

Euler's formula is a stunning revelation within mathematics, connecting complex numbers and exponential functions beautifully. Expressed as:

• \(e^{i\theta} = \cos \theta + i\sin \theta\)

Euler's formula demonstrates that complex exponentials can encode both the direction (argument \(\theta\)) and magnitude.
The formula naturally extends to the representation of any complex number in polar form. If we have a complex number \(z = re^{i\theta}\), \(r\) is the modulus \(|z|\), and \(\theta\) is the argument \(\arg(z)\). Therefore, Euler's formula provides an elegant description of complex numbers, allowing them to be manipulated more easily

• Example: Express \(z = 1 + i\) in polar form as \(\sqrt{2}e^{i \frac{\pi}{4}}\).

This translation into polar form is advantageous for tasks including calculating powers or roots of complex numbers.
Euler's formula proves that complex numbers are not just algebraic abstractions but are deeply woven into the fabric of mathematics, providing solutions and insights across wide-ranging disciplines.