Question

Find each of the following in rectangular form \(x+i y\) and check your results by computer. Remember to save time by doing as much as you can in your head. $$e^{(i \pi / 4)+(\ln 2) / 2}$$

Short answer

The rectangular form is \(1 + i\).

Step-by-step solution

Simplify the Exponent

The given expression is \(e^{(i \pi / 4) + (\ln 2) / 2}\). Start by using the property of exponents: \(e^{a+b} = e^a \, e^b\). So we separate it into two parts: \(e^{i \pi / 4}\) and \(e^{(\ln 2) / 2}\).

Evaluate \(e^{i \pi / 4}\)

Using Euler's formula, \(e^{i\theta} = \cos \theta + i \sin \theta\), we get \[ e^{i \pi / 4} = \cos(\pi / 4) + i \sin(\pi / 4) = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \].

Evaluate \(e^{(\ln 2) / 2}\)

By definition, \(e^{\ln x} = x\). Therefore, we have \(e^{(\ln 2) / 2} = 2^{1/2} = \sqrt{2}\).

Combine the Two Results

Now, multiply the results from Steps 2 and 3: \[ \sqrt{2} \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = 1 + i. \]

Key concepts

Rectangular Form

The rectangular form of a complex number is expressed as \( x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. To convert an expression into rectangular form, separate the real and imaginary components.
In the solution exercise, we ended up with \( e^{(i \,\pi / 4) + (\ln 2) / 2} \). By working through several steps, we separated it into two parts, evaluated them, and combined them to achieve the rectangular form of \( 1 + i \).
Being comfortable with converting complex expressions into this form simplifies many tasks in mathematics and engineering!

Euler's Formula

Euler's formula, given by \( e^{i\theta} = \cos \theta + i\sin \theta \), is fundamental in converting complex exponentials into trigonometric form. This formula bridges exponential functions and trigonometric functions.
In the exercise, we used Euler’s formula to evaluate \( e^{i\pi/4} \) as \( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \). Understanding this fundamental relationship is crucial when working with complex numbers and simplifies conversions between exponential and rectangular forms.

Exponential Functions

Exponential functions often appear in the form \( e^{x} \), where \( e \) is the base of natural logarithms, and \( x \) is the exponent. When dealing with complex numbers, exponential functions can be broken down using properties such as \( e^{a+b} = e^{a} \, e^{b} \).
In the exercise, we applied this property: \( e^{(i \pi/4) + (\ln 2)/2} = e^{i\pi/4} \cdot e^{(\ln 2)/2} \), simplifying our problem into manageable parts. Recognizing these properties can make complex calculations clearer and more intuitive.

Logarithms

Logarithms, particularly natural logarithms (logarithms with base \( e \)), play an important role in simplifying and solving exponential expressions. The natural logarithm of a number \( x \) is denoted as \( \ln(x) \), and it has the property that \( e^{\ln(x)} = x \).
In the exercise, we handled \( e^{(\ln 2) / 2} \) by realizing that it equates to \( \sqrt{2} \). Recognizing and utilizing logarithmic properties is key to transforming and simplifying complex expressions.

Trigonometric Identities

Trigonometric identities are vital tools for simplifying expressions involving trigonometric functions. Key identities include \( \cos(\theta) \) and \( \sin(\theta) \), which appear frequently when dealing with Euler's formula.
In our example, knowing the values \( \cos(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2} \) allowed us to transform \( e^{i\pi/4} \) directly. Trigonometric identities help simplify complex calculations and convert expressions into more workable forms.