Question

use Euler’s formula to write the given expression in the form a + ib. $$ \exp (1+2 i) $$

Short answer

Question: Rewrite the expression \(\exp(1+2i)\) in the form a + ib using Euler's formula. Answer: The expression \(\exp(1+2i)\) can be rewritten in the form a + ib as \((\exp(1) \cos(2)) + i (\exp(1) \sin(2))\).

Step-by-step solution

Separate exponent into real and imaginary parts

Let's break down the expression \(\exp(1+2i)\) into the sum of its real and imaginary exponent parts: $$ \exp(1 + 2i) = \exp(1) \cdot \exp(2i) $$ Here, \(\exp(1)\) represents the real part and \(\exp(2i)\) represents the imaginary part.

Apply Euler's formula to the imaginary part

Now, we will apply Euler's formula to the imaginary part of the expression, \(\exp(2i)\): $$ e^{2i} = \cos(2) + i \sin(2) $$ Here, \(\cos(2)\) and \(\sin(2)\) represent real numbers.

Write the final expression

Now, we can write the final expression by substituting the Euler's formula in the expression from Step 1: $$ \exp(1) \cdot \exp(2i) = \exp(1) (\cos(2) + i \sin(2)) $$ The given expression in the form a + ib is: $$ a + ib = (\exp(1) \cos(2)) + i (\exp(1) \sin(2)) $$

Key concepts

Complex Exponentials

Understanding complex exponentials is pivotal in areas like electrical engineering and quantum mechanics. To begin with, a complex exponential is an expression of the form \( e^{i\theta} \), where \( e \) is the base of the natural logarithm, and \( \theta \) is a real number representing an angle in radians. How this translates to complex numbers is beautifully captured by Euler's formula, which states that
\[ e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta) \].
This intriguing result links the exponential function with trigonometric functions of sine and cosine, transforming the complex exponential into a form that incorporates both the real and the imaginary parts. Applying Euler's formula, as seen in the exercise, allows us to express complex exponentials in a more interpretable form known as the form \( a + ib \), where \( a \) is the real part and \( b \) is the imaginary part.

Complex Numbers in Trigonometric Form

Complex numbers can be daunting, but their trigonometric form makes them much more approachable. The general form of a complex number is \( a + ib \), with \( a \) and \( b \) being real numbers and \( i \) representing the imaginary unit. Now, representing a complex number in trigonometric form involves expressing it as \( r(\text{cos}(\theta) + i\text{sin}(\theta)) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument of the complex number—which is just the angle formed by the line representing the complex number on the complex plane and the positive direction of the real axis.
This form is essentially derived from Euler's formula and is vital because it offers an intuitive geometric interpretation of complex numbers. It also simplifies the multiplication and division of complex numbers since you can directly multiply or divide their magnitudes (moduli) and add or subtract their arguments (angles). In the step-by-step solution from the exercise, we've utilized this trigonometric form to express \( \text{exp}(2i) \) as part of the final answer.

Exponential Functions

Exponential functions, denoted as \( e^{x} \), are fundamental in various scientific fields, defining growth or decay processes that are constantly proportional to the current value.
In our exercise, we come across the term \( \text{exp}(1) \), which is another notation for \( e \) raised to the power of the real number 1.
When dealing with exponential functions with complex exponents, the behavior is different but related to real exponentials. The function \( e^{x} \) is known for its property of having a rate of change proportional to its value for any real \( x \) which leads to its distinctive smooth curve on a graph. In the context of complex numbers, we still deal with this growth aspect but we project it onto the two-dimensional plane where it manifests in the form of rotations and dilations, thanks to Euler's formula associating complex exponentials with sinusoids. This dual nature of exponentials is key to decoding complex signals in practical applications such as signal processing.