Question

Find the norm of each of the following functions on the given interval and state the normalized function \(\cos n x\) on \((0, \pi)\)

Short answer

The norm of \( \cos(nx) \) on \( (0, \pi) \) is \( \frac{\sqrt{2\pi}}{2} \). The normalized function is \( f_{normalized}(x) = \frac{ 2 \cos(nx) }{ \sqrt{2\pi} } \).

Step-by-step solution

Understand the Problem

You need to find the norm of the function \(\cos(nx)\) over the interval \( (0, \pi) \). Additionally, you need to state the normalized function.

Definition of Norm for Functions

For a function \( f(x) \) defined on the interval \( (a, b) \), the norm is defined as \[ ||f|| = \sqrt{ \int_a^b |f(x)|^2 dx }. \]

Set Up the Integral

Substitute \( f(x) = \cos(nx) \), \( a = 0 \) and \( b = \pi \) in the norm definition: \[ || \cos(nx) || = \sqrt{ \int_0^\pi |\cos(nx)|^2 dx }. \]

Simplify the Integral

Use the fact that \[ |\cos(nx)|^2 = ( \cos(nx) )^2 \]. So, the integral becomes: \[ || \cos(nx) || = \sqrt{ \int_0^\pi ( \cos(nx) )^2 dx }. \]

Evaluate the Integral

The integral can be evaluated using the identity \(( \cos(nx) )^2 = \frac{1 + \cos(2nx)}{2} \): \[( || \cos(nx) || )^2 = \int_0^\pi \frac{1 + \cos(2nx)}{2} dx .\] This splits into two integrals: \[( || \cos(nx) || )^2 = \frac{1}{2} \int_0^\pi 1 \space dx + \frac{1}{2} \int_0^\pi \cos(2nx) \space dx. \]

Solve the Integrals

Evaluate each part separately: \[ \frac{1}{2} \int_0^\pi 1 \space dx = \frac{1}{2} \cdot \pi = \frac{\pi}{2} \] and \[\frac{1}{2} \int_0^\pi \cos(2nx) \space dx = 0 \] since the integral of \cos(2nx) over a full period is zero.

Compute the Norm

Combine the results to get: \[( || \cos(nx) || )^2 = \frac{\pi}{2}. \] Hence, \[( || \cos(nx) || ) = \sqrt{ \frac{\pi}{2} } = \frac{\sqrt{\pi}}{\sqrt{2}} = \frac{\sqrt{2\pi}}{2}. \]

Normalize the Function

The normalized function is given by dividing \cos(nx)\ by its norm: \[ f_{normalized}(x) = \frac{ \cos(nx) }{ || \cos(nx) || } = \frac{ \cos(nx) }{ \frac{\sqrt{2\pi}}{2} } = \frac{ 2 \cos(nx) }{ \sqrt{2\pi} }. \]

Key concepts

Norm

In mathematics, especially in the context of function spaces, the norm provides a measure of the 'size' or 'length' of a function. For a function defined on a specific interval, the norm can be thought of as an extension of the Euclidean length to function spaces. The norm of a function \( f(x) \) is generally given by the formula:
\[ ||f|| = \sqrt{ \int_a^b |f(x)|^2 dx }. \]
This formula effectively computes the area under the curve of \( |f(x)|^2 \) and then takes the square root of this area. The squared term ensures that the norm is always non-negative, and the integral ensures that the entire interval \( (a, b) \) is considered.
In our specific problem, we examined the function \( \cos(nx) \) over the interval \( (0, \pi) \). By substituting \( f(x) = \cos(nx) \), \( a = 0 \), and \( b = \pi \), we set up the integral:
\[ || \cos(nx) || = \sqrt{ \int_0^\pi ( \cos(nx) )^2 dx }. \]

Integral Calculus

Integral calculus deals with the concept of finding integrals, which can be thought of as the area under a curve. In the context of finding the norm, integral calculus helps in calculating the area under the curve of \( |f(x)|^2 \) over the given interval. To solve the integral \( \sqrt{ \int_0^\pi ( \cos(nx) )^2 dx } \), we used a trigonometric identity to simplify the integrand.
The identity \( ( \cos(nx) )^2 = \frac{1 + \cos(2nx)}{2} \) transforms the integral into a more manageable form:
\[ \sqrt{ \int_0^\pi \frac{1 + \cos(2nx)}{2} dx }. \]
This splits into two simple integrals:
\[ \frac{1}{2} \int_0^\pi 1 \, dx + \frac{1}{2} \int_0^\pi \cos(2nx) \, dx. \]
By evaluating these integrals separately, we find that the integral of a constant over \( (0, \pi) \) is \( \frac{\pi}{2} \), and the integral of \( \cos(2nx) \) over a full period cancels out to zero, leading to the result:
\[ ( || \cos(nx) || )^2 = \frac{\pi}{2}. \]

Normalization

Normalization in the context of functions means adjusting the function so that its norm equals 1. This is crucial when comparing functions of different magnitudes or when ensuring functions conform to certain standards, such as in quantum mechanics.
In our exercise, after finding the norm of \( \cos(nx) \) as \( \frac{\sqrt{2\pi}}{2} \), we normalized \( \cos(nx) \) by dividing it by its norm:
\[ f_{normalized}(x) = \frac{ \cos(nx) }{ || \cos(nx) || } = \frac{ \cos(nx) }{ \frac{\sqrt{2\pi}}{2} } = \frac{ 2 \cos(nx) }{ \sqrt{2\pi} }. \]
This results in a function whose norm is now 1, making it easier to work with and compare to other normalized functions. Normalization plays a key role in various scientific and mathematical fields where the relative sizes of functions are essential.