Problem 36 - Essential Calculus

Question

Use a power series and the fact that \(i^{2}=-1\) to verify the formula. $$ g(x)=\frac{1}{2}\left(e^{i x}+e^{-i x}\right)=\cos x $$

Step-by-Step Solution

Answer

\(\frac{1}{2} \left(e^{i x}+e^{-i x}\right) = \cos x\). The given formula is verified through the use of the power series of the exponential function and the property of \(i^{2} = -1\).

Step 1: Write the power series representation of \(e^{ix}\) and \(e^{-ix}\)

Power series representation of an exponential function \(e^{u}\) is given by \(\sum_{n=0}^{\infty}\frac{u^{n}}{n!}\). So we can write \(e^{ix}\) and \(e^{-ix}\) as power series: \(e^{ix}=\sum_{n=0}^{\infty}\frac{(ix)^{n}}{n!}\) and \(e^{-ix}=\sum_{n=0}^{\infty}\frac{(-ix)^{n}}{n!}\).

Step 2: Break down the power series

Now break down these series by separating the even and odd powers terms. For \(e^{ix}\), the power series becomes \(\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!} + i \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\). Similarly for \(e^{-ix}\), the power series becomes \(\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!} - i \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\). Note here that \(\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\) is the power series for \(\cos x\) and \(\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\) is the power series for \(\sin x\).

Step 3: Add the two series

Now we add the two series \(e^{ix}\) and \(e^{-ix}\). All the imaginary parts cancel out and we get \(\frac{1}{2} \left(e^{i x}+e^{-i x}\right) = \frac{1}{2} \left(2 \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\right) = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\). This series is the power series representation of \(\cos x\).

Key Concepts

Power Series

A power series is a way to express functions as infinite sums of terms. It can be thought of like an infinite polynomial. Each term is based on raising a variable to a power, multiplied by a coefficient. Power series are particularly useful in calculating functions that are typically difficult to compute exactly.

In mathematics, the power series representation of a well-known function like the exponential function is given by \[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.\]
This means that we can write any exponential function, such as \(e^{ix}\) and \(e^{-ix}\), as infinite power series. The variable \(x\) is replaced by \(ix\) or \(-ix\) in each term and expanded:

• \(e^{ix} = \sum_{n=0}^{\infty} \frac{(ix)^n}{n!}\)
• \(e^{-ix} = \sum_{n=0}^{\infty} \frac{(-ix)^n}{n!}\)

These expansions allow us to analyze functions even when they involve complex numbers. They help us express functions that can otherwise be cumbersome to deal with directly.

Complex Numbers

Complex numbers extend the idea of the standard real number system, incorporating an imaginary unit \(i\), where \(i^2=-1\). This complex unit is essential for a wide array of mathematical calculations and applications.

When dealing with terms like \(e^{ix}\), complex numbers come into play. Here, the expression includes the imaginary unit multiplied by a real number. Complex numbers can take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the square root of -1. These numbers are particularly useful when working with wave functions and oscillations in fields such as engineering and physics.

Using power series expansions with complex numbers involves handling terms where \(i\) and real numbers interact:

• The powers of \(i\) cycle in a repetitive manner: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then repeating.
• This property helps in simplifying series solutions and finding real and imaginary components separately.

Understanding complex numbers is crucial for creating and simplifying equations that involve trigonometric identities such as cosine functions, incorporating exponential terms with imaginary components.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It describes the horizontal coordinate of a point on the unit circle as the circle is traversed. More formally, it is often written as \(\cos x\). Cosine is pivotal in a variety of applications, from describing light waves to engineering signals.

Its power series representation is essential for computing cosine values without directly calculating angles. The series expansion for the cosine function looks like this:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.\]Each term in this series is obtained by alternating between adding and subtracting even-powered terms of \(x\), divided by the factorial of even integers. When dealing with complex exponentials like in the exercise \(\frac{1}{2}(e^{ix} + e^{-ix})\), it elegantly results in the cosine series.
The beauty of this method is how it simplifies computations using imaginary numbers to eventually arrive back at a real, tangible function like cosine. This approach is widely used in fields like signal processing, where calculating precise waveforms is critical.