Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the following. (a) \(e^{2 \pi i}\) (b) \(e^{-\pi i}\)

Step-by-Step Solution

Answer

(a) \(e^{2 \pi i} = 1\), (b) \(e^{-\pi i} = -1\)

Step 1: Apply Euler's formula for \(e^{2 \pi i}\)

Using Euler's formula, substitute \(x = 2 \pi\) into \(e^{ix} = \cos(x) + i \sin(x)\) to obtain \(e^{2 \pi i} = \cos(2 \pi) + i \sin(2 \pi)\). Now compute the values of \(\cos(2 \pi)\) and \(\sin(2 \pi)\) and substitute them back into the equation.

Step 2: Simplify \(e^{2 \pi i}\)

Since \(\cos(2 \pi) = 1\) and \(\sin(2 \pi) = 0\), we find \(e^{2 \pi i} = 1 + i*0\), which simplifies to \(e^{2 \pi i} = 1\).

Step 3: Apply Euler's formula for \(e^{-\pi i}\)

Again using Euler's formula, substitute \(x = -\pi\) into \(e^{ix} = \cos(x) + i \sin(x)\) to obtain \(e^{- \pi i} = \cos(-\pi) + i \sin(-\pi)\). Compute the values of \(\cos(-\pi)\) and \(\sin(-\pi)\) and put them back into the equation.

Step 4: Simplify \(e^{-\pi i}\)

Since \(\cos(-\pi) = -1\) and \(\sin(-\pi) = 0\), we find \(e^{- \pi i} = -1 + i*0\), which simplifies to \(e^{-\pi i} = -1\).

Key Concepts

Complex numbers

Complex numbers are a type of number that extend the idea of the one-dimensional number line to a two-dimensional complex plane by including the square root of -1, which is represented as the imaginary unit 'i'. A complex number is usually written in the form of 'a + bi', where 'a' is the real part, and 'bi' is the imaginary part.

Understanding complex numbers is crucial for solving a variety of problems in mathematics and physics. They allow us to perform calculations that involve square roots of negative numbers, and they are essential in the study of waves, quantum mechanics, and electrical engineering.

Real-world applications involve solving polynomial equations, where the roots may be complex, and they are crucial in the field of complex analysis, which is the study of functions that operate on complex numbers.

Exponential functions

Exponential functions are a fundamental class of functions that appear frequently in mathematics, characterized by constant ratios in successive terms of a geometric sequence. In its simplest form, an exponential function is written as 'b^x', where 'b' is the base and 'x' is the exponent. When the base is Euler's number 'e', approximately equal to 2.71828, we get the most commonly used exponential function in mathematics, denoted as 'e^x'.

This function has distinctive properties such as the slope of the curve 'e^x' being equal to its value at any point, which is a distinctive and unique property that makes it useful for modeling growth and decay processes such as compound interest, population growth, and radioactive decay. In calculus, the exponential function is also used to define the exponential map and is closely related to the concept of logarithms.

Trigonometric functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), and they are crucial for studying periodic phenomena, such as waves and oscillations.

The major importance of trigonometric functions lies in their ability to describe the properties of any angle, not just those in right triangles. They are applicable in various branches of science and engineering, including acoustics, electronics, and even celestial navigation. Trigonometry holds a significant place in geometric calculations, helping to solve problems involving triangles, circles, oscillations, and waves.