Problem 7 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=1.5+2 i $$

Step-by-Step Solution

Answer

Recall Euler's Formula to convert \( e^z \) into \( a+ib \) for complex \( z \).

Step 1: Recall Euler's Formula

To express \( e^z \) in the form \( a + ib \) for a complex number \( z = x + yi \), use Euler's formula: \( e^{z} = e^{x} (\cos y + i \sin y) \).

Key Concepts

Euler's formula

Euler's formula is a beautiful and revolutionary finding in mathematics. It bridges the field of exponential functions with trigonometry. The formula is written as \( e^{ix} = \cos x + i \sin x \). This means that you can represent complex numbers in an effortless form using just one exponential term and trigonometric functions of real angles.

Understanding Euler's formula means understanding that:\

• \( e^{ix} \) explains the relationship between exponential growth and circular motion.
• It shows how complex exponential functions curve naturally into the circle of trigonometric angles.

This formula is fundamental for converting complex numbers between different representations. Whether it's changing from rectangular to polar or vice versa, Euler's formula is key to making those transformations smooth and intuitive.

Exponential form of complex numbers

To explore the exponential form of complex numbers, imagine replacing angles and magnitudes with exponential operations. This form is expressed as \( z = re^{i\theta} \). Here, \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the angle (or argument) it makes with the positive real axis.

Here’s how to break it down:

• \( r = |z| \) is calculated as \( \sqrt{x^2 + y^2} \), which is the absolute value of the complex number \( x + yi \).
• \( \theta = \arg(z) \) represents the angle, typically using \( \tan^{-1}(y/x) \).

With the exponential form, complex numbers can easily undergo mathematical operations like multiplication and division. This is because multiplying two complex numbers in exponential form involves simply multiplying their magnitudes \( r_1r_2 \) and adding their angles \( \theta_1 + \theta_2 \). This method simplifies your calculations for powers and roots in complex number operations.

Trigonometric form of complex numbers

The trigonometric form provides another way to express complex numbers. It resembles directly the idea of plotting points on a circular path. In this form, a complex number \( z = x + yi \) is written as \( z = r(\cos \theta + i\sin \theta) \). This is sometimes also referred to as the polar form of a complex number.

Let's understand its components:

• The modulus \( r \) is the distance from the origin to the point, calculated using \( \sqrt{x^2 + y^2} \).
• The argument \( \theta \) signifies the angle the line from the origin to the point makes with the positive real axis.

The trigonometric form of complex numbers is particularly useful when visualizing or graphing these numbers. For conversions, this form beautifully shows how any complex number relates to a flight around a circle. With this understanding:

• Complicated operations like multiplication and raising to powers become easier.
• Visualizing transformations such as rotation on the complex plane become almost intuitive.