Question

From the Euler relations, deduce that: (a) \(e^{-i \pi}=-1\) (c) \(e^{i \pi / 4}=\frac{\sqrt{2}}{2}(1+i)\) (b) \(e^{i \pi / 3}=\frac{1}{2}(1+\sqrt{3} i)\) \((d) e^{-3 i \pi / 4}=-\frac{\sqrt{2}}{2}(1+i)\)

Short answer

(a) Euler's formula gives \( e^{-i \pi} = -1 \); (c) \( e^{i \pi / 4} = \frac{\sqrt{2}}{2}(1+i) \); (b) \( e^{i \pi / 3} = \frac{1}{2}(1+\sqrt{3} i) \); (d) \( e^{-3 i \pi / 4} = -\frac{\sqrt{2}}{2}(1+i) \).

Step-by-step solution

Understanding Euler's Formula

Euler's formula states that for any real number \( \theta \), \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). This formula is essential for solving these types of problems involving complex exponentials.

Solve for (a) \( e^{-i\pi} \)

Using Euler's formula, set \( \theta = -\pi \): \( e^{-i\pi} = \cos(-\pi) + i\sin(-\pi) \). We know \( \cos(-\pi) = -1 \) and \( \sin(-\pi) = 0 \). Thus, \( e^{-i\pi} = -1 + 0i = -1 \).

Solve for (c) \( e^{i \pi / 4} \)

Set \( \theta = \pi/4 \) in Euler’s formula: \( e^{i \pi / 4} = \cos(\pi/4) + i\sin(\pi/4) \). We know \( \cos(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2} \). Thus, \( e^{i \pi / 4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}(1+i) \).

Solve for (b) \( e^{i \pi / 3} \)

Set \( \theta = \pi/3 \) in Euler’s formula: \( e^{i \pi / 3} = \cos(\pi/3) + i\sin(\pi/3) \). Here, \( \cos(\pi/3) = \frac{1}{2} \) and \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \). Thus, \( e^{i \pi / 3} = \frac{1}{2} + i\frac{\sqrt{3}}{2} = \frac{1}{2}(1 + \sqrt{3}i) \).

Solve for (d) \( e^{-3i\pi/4} \)

Set \( \theta = -3\pi/4 \) in Euler’s formula: \( e^{-3i\pi/4} = \cos(-3\pi/4) + i\sin(-3\pi/4) \). From trigonometric identities, \( \cos(-3\pi/4) = -\frac{\sqrt{2}}{2} \) and \( \sin(-3\pi/4) = -\frac{\sqrt{2}}{2} \). Thus, \( e^{-3i\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}(1+i) \).

Key concepts

Complex Exponentials

Complex exponentials are a captivating concept in mathematics, combining the worlds of exponential functions and complex numbers. When you raise the number "e" (approximately 2.71828) to a complex power, you get a complex exponential function. This is expressed as:
\( e^{i\theta} \), where \( i \) is the imaginary unit, and \( \theta \) is the angle in radians. The beauty of complex exponentials is that they encapsulate a rotating movement on the complex plane. This rotation is made evident when analyzing Euler's formula, which shows how these exponential expressions can relate angle and sinusoidal functions.
Understanding complex exponentials is key to solving problems involving waveforms or oscillations, such as those in electrical engineering, quantum physics, and signal processing. It unlocks a sophisticated but elegantly simple way to handle periodic functions and rotations.

Trigonometric Identities

Trigonometric identities provide crucial relationships between the angles and sides of a triangle. These identities help simplify expressions and enable the solving of complex mathematical problems involving angles. The most commonly used trigonometric identities include the sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)) functions, among others.

• The Pythagorean identity: \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
• The angle sum and difference identities, such as \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)
• The double-angle identities, like \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

These identities are crucial when applying Euler’s formula, allowing us to express complex numbers in exponential form while also handling terms like \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). Mastery of trigonometric identities facilitates understanding complex exponentials and decoding Euler’s magic.

Euler Relations

Euler Relations, rooted in Euler's formula, connect various mathematical worlds: exponential functions, trigonometry, and complex numbers. The primary form of this relation is:
\( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).
Euler's Relations allow for the transformation of complex exponentials into trigonometric form and vice versa. This transformation simplifies analysis and computations in fields like electrical engineering, where sinusoidal waves are prevalent.
Euler's relations also underpin the famous identity:

• \( e^{i\pi} + 1 = 0 \), which elegantly ties together the five most important numbers in mathematics: e, i, pi, 1, and 0.

By mastering Euler Relations, you gain the power to handle complex analysis involving rotations and oscillations.

Complex Numbers

Complex numbers expand the idea of numbers beyond the real line into a plane, combining a real part and an imaginary part. A complex number could be written as \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
These numbers allow the extension of real-number theories with added dimensions, making them fundamental in solving problems where square roots of negative numbers appear. The geometric interpretation of complex numbers places them on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.
Complex numbers facilitate the representation of waves, rotations, and other periodic phenomena in a more comprehensive manner. They are indispensable in engineering, physics, and applied mathematics, providing a robust framework for modeling real-world behaviors.