Problem 31 - Advanced Engineering Mathematics

Question

In Problems \(29-34\), express \(\operatorname{Ln} z\) in the form \(a+i b\). $$ z=-12+5 i $$

Step-by-Step Solution

Answer

\( \operatorname{Ln} z = \ln(13) + i\left( \pi + \arctan(-5/12) \right) \).

Step 1: Identify the components of the complex number

The complex number given is \( z = -12 + 5i \). This has a real part \( x = -12 \) and an imaginary part \( y = 5 \).

Step 2: Calculate the magnitude of z

The magnitude of a complex number \( z = x + yi \) is given by the formula \( r = \sqrt{x^2 + y^2} \). Applying this formula, we have \( r = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \).

Step 3: Calculate the argument of z

The argument \( \theta \) of a complex number \( z = x + yi \) is calculated using \( \tan \theta = \frac{y}{x} \). Thus, \( \theta = \arctan \left( \frac{5}{-12} \right) \). Since the complex number is in the second quadrant, we adjust by adding \( \pi \): \( \theta = \pi + \arctan \left( -\frac{5}{12} \right) \).

Step 4: Write the formula for the natural logarithm of z

The natural logarithm of a complex number \( z = re^{i\theta} \) is given by \( \operatorname{Ln} z = \ln r + i\theta \).

Step 5: Substitute magnitude and argument into the logarithm expression

Using the results from the previous steps, substitute into the formula: \( \operatorname{Ln} z = \ln(13) + i\left( \pi + \arctan \left( -\frac{5}{12} \right) \right) \).

Step 6: Express in the form a + ib

Hence, in the form \( a + ib \), we have \( \operatorname{Ln} z = \ln(13) + i\left( \pi + \arctan \left( -\frac{5}{12} \right) \right) \).

Key Concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. The standard form of a complex number is expressed as \( z = x + yi \), where:

• \( x \) is the real part.
• \( y \) is the imaginary part, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \).

Complex numbers are essential in mathematics because they allow us to solve equations that do not have real number solutions. For instance, the equation \( x^2 + 1 = 0 \) has no real solutions, but in the world of complex numbers, it does: \( x = i \) and \( x = -i \).
When graphed on the complex plane, the real part \( x \) is plotted on the horizontal axis, and the imaginary part \( yi \) is plotted on the vertical axis. This visualization enables many applications, from engineering to physics, making complex numbers a fundamental concept in advanced mathematics.

Magnitude of a Complex Number

The magnitude (or modulus) of a complex number provides the straight-line distance from the origin (0,0) to the point \( z = x + yi \) on the complex plane. This is similar to finding the hypotenuse of a right triangle.
The formula for calculating the magnitude \( r \) is:

• \( r = \sqrt{x^2 + y^2} \)

In this exercise, for \( z = -12 + 5i \), the magnitude is found by substituting the values \( x = -12 \) and \( y = 5 \) into the formula, yielding \( r = \sqrt{(-12)^2 + 5^2} = \sqrt{169} = 13 \).

Remember, the magnitude is always a non-negative number and represents the "size" or "length" of the complex number, without considering its direction.

Argument of a Complex Number

The argument of a complex number is the angle \( \theta \) that the line connecting \( z = x + yi \) to the origin makes with the positive real axis. It essentially tells us the direction of the complex number.
Calculating the argument typically involves using the formula:

• \( \tan \theta = \frac{y}{x} \)

For \( z = -12 + 5i \), the calculation is \( \theta = \arctan \left( \frac{5}{-12} \right) \). Because this complex number lies in the second quadrant (where the real part is negative and the imaginary part is positive), we adjust the angle by adding \( \pi \):

• \( \theta = \pi + \arctan \left(- \frac{5}{12} \right) \).

This correction ensures that the angle accurately reflects the complex number's position in the complex plane.

Natural Logarithm

The natural logarithm of a complex number is an extension of the logarithm concept from real numbers to complex numbers. It is defined for a complex number \( z = re^{i\theta} \) as:

• \( \operatorname{Ln} z = \ln r + i\theta \)

The natural logarithm here involves:

• \( \ln r \): the natural logarithm of the magnitude.
• \( i\theta \): the product of the imaginary unit and the argument \( \theta \).

For \( z = -12 + 5i \) in the exercise, substitute \( r = 13 \) and the calculated \( \theta \) into the formula. This results in \( \operatorname{Ln} z = \ln(13) + i\left( \pi + \arctan \left( -\frac{5}{12} \right) \right) \).
This expression in the form \( a + ib \) illustrates how complex logarithms combine the magnitude and direction of a complex number into a singular value, offering insights into both its size and orientation.