Problem 7 - Advanced Engineering Mathematics

Question

Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each function in the set. $$ \\{\sin x, \sin 3 x, \sin 5 x, \ldots\\} ; \quad[0, \pi / 2] $$

Step-by-Step Solution

Answer

The functions are orthogonal on \([0, \pi/2]\); each has a norm of \(\frac{\sqrt{\pi}}{2}\).

Step 1: Define Orthogonality

Two functions \( f(x) \) and \( g(x) \) are orthogonal on an interval \([a, b]\) if the integral of their product over this interval is zero. Mathematically, \(\int_a^b f(x)g(x) \, dx = 0\). In this problem, we need to show that \( \sin nx \) and \( \sin mx \) are orthogonal for \(n eq m\) on \([0, \pi/2]\).

Step 2: Compute the Inner Product

The inner product of \( \sin nx \) and \( \sin mx \) on \([0, \pi/2]\) is given by \(\int_0^{\pi/2} \sin nx \cdot \sin mx \, dx\). Use the identity \( \sin nx \sin mx = \frac{1}{2} [\cos(n-m)x - \cos(n+m)x] \) to simplify the expression.

Step 3: Evaluate the Integral

Calculate \[\int_0^{\pi/2} \frac{1}{2} [\cos(n-m)x - \cos(n+m)x] \, dx\]. This separates into two integrals: \( \frac{1}{2} \int_0^{\pi/2} \cos(n-m)x \, dx \) and \( -\frac{1}{2} \int_0^{\pi/2} \cos(n+m)x \, dx \). Each of these results in \( \frac{1}{2} \left[ \frac{\sin(n-m)x}{n-m} \right]_0^{\pi/2} \) and \( -\frac{1}{2} \left[ \frac{\sin(n+m)x}{n+m} \right]_0^{\pi/2} \), which both equal zero as \( n-m eq 0 \) and \( n+m eq 0 \).

Step 4: Conclude Orthogonality

Since the integral evaluates to zero for \(n eq m\), the functions \( \sin nx \) and \( \sin mx \) are orthogonal on \([0, \pi/2]\). This demonstrates that each pair of distinct functions in the given set is orthogonal over the specified interval.

Step 5: Define Norm of a Function

The norm of a function \( f(x) \) on \([a, b]\) is defined as \( \sqrt{\int_a^b [f(x)]^2 \, dx} \). For \( \sin nx \), calculate \( \int_0^{\pi/2} \sin^2(nx) \, dx \) using the identity \( \sin^2(nx) = \frac{1}{2}[1 - \cos(2nx)] \).

Step 6: Calculate Integral for Norm

Compute \(\int_0^{\pi/2} \frac{1}{2}[1 - \cos(2nx)] \, dx = \frac{1}{2} \left[ x - \frac{\sin(2nx)}{2n} \right]_0^{\pi/2} \). This results in \(\frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4}\).

Step 7: Determine the Norm

The norm of \( \sin nx \) is \( \sqrt{\frac{\pi}{4}} = \frac{\sqrt{\pi}}{2} \). Thus, each function in the set \( \{ \sin x, \sin 3x, \sin 5x, \ldots \} \) has this norm on \([0, \pi/2]\).

Key Concepts

Inner Product

The concept of an inner product is central to understanding orthogonal functions. In simple terms, the inner product of two functions is a way to multiply these functions, similar to how we multiply numbers. For two functions, \( f(x) \) and \( g(x) \), defined over an interval \([a, b]\), the inner product is calculated as \( \int_a^b f(x) g(x) \, dx \). This integral effectively "measures" how much the two functions overlap over the interval.
This concept is crucial because if two functions are orthogonal, their inner product will be zero. This means there is no overlap between them within the specified interval. In the exercise, we evaluated the inner product of \( \sin nx \) and \( \sin mx \) over the interval \([0, \pi/2]\) to show that they are orthogonal (or perpendicular in higher-dimensional space) when \( n eq m \). Orthogonality in this context means that the sine waves of different frequencies (such as \(\sin x \) and \(\sin 3x \)) do not interfere constructively over the given interval.

Fourier Series

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. These series are powerful tools for analyzing and representing periodic functions. When we want to approximate a function using trigonometric functions, we use the Fourier series to express it as a sum of orthogonal sines and cosines.
For the problem at hand, each sine function \( \sin nx \) (where \( n \) is an odd integer) can be considered part of a potential Fourier series representation of a function defined over \([0, \pi/2]\). The orthogonality of the sine functions ensures that each component of the series captures a unique part of the wave, without redundancy, making the Fourier series an efficient and unique representation of the function across the specified interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equation are defined. These identities are essential for simplifying the product of trigonometric functions, especially in inner product calculations.
The problem uses the identity \( \sin nx \sin mx = \frac{1}{2} [\cos(n-m)x - \cos(n+m)x] \) to simplify the product of sine functions. This identity allows us to express the product of two sine waves as a sum of cosine waves, enabling easier integration over the interval \([0, \pi/2]\).
Other important identities are those related to the squares of sine and cosine functions. For instance, \( \sin^2(nx) = \frac{1}{2}[1 - \cos(2nx)] \) helps in calculating the norm of the functions since it simplifies the process of finding their average value over one period. Understanding and applying these identities correctly is fundamental for dealing with questions involving orthogonality and norms of trigonometric functions.