Orthogonality

In order to derive some of the results included in this section, we make use of the product-to-sum trigonometric identities, given as follows:

Theorem: Orthogonality

Two nonzero functions, $f(x)$ and $g(x)$ are said to be orthogonal on $[a, b]$ if their inner product is given by

We are more interested in the so-called orthogonality relations a set of nonzero functions have: a sequence of functions $\left\{f_{i}(x)\right\}$ is said to be $m u$ tually orthogonal on $[a, b]$ if,

Next, we prove the orthogonality relations given by Theorem 1.1 for the set of functions $\{\sin (n x), \sin (m x)\}$. We want to prove that they are mutually orthogonal on $[-\pi, \pi]$; to do that we will make use of the appropriate identity from (1.7a)-(1.7c) and relevant double angle formula relations. We start with the integral on the LHS of Eq. (1.9) where, here, $f_{i}(x)=\sin (n x)$ and $f_{j}(x)=$ $\sin (m x)$ and $a=-\pi, b=\pi$ :

We need to consider two cases separately for (i) $m=n$ and (ii) $m \neq n$. For the case where $m=n$, Eq. (1.10) becomes:

note that in going from left to right in Eq. (1.11), we made use of Eq. (1.5) since the function $\sin ^{2}(n x)$ is an even function. To integrate the RHS of Eq. (1.11), we make use of the double angle formula $\cos (2 n x)=1-\sin ^{2}(n x)$. This gives:

Next, for the case $m \neq n$, we need to make use of the sum-to-product identity given by Eq. (1.7c).

Note that the integrand is the product of two odd functions, $\sin (n x) \sin (m x)$ which gives an even function. Integrating Eq. (1.14) with respect to $x$, gives

Notice that in Eq. (1.15), we made use of Eq. (1.5) since we are integrating an even function from $-\pi$ to $\pi$. Firstly, we know that $m \neq n$ and therefore the denominator is well-defined. Secondly, observe that $m$ and $n$ are integers and hence $m-n$ and $m+n$ are also integers. It follows that $\sin [(m-n) x]$ and $\sin [(n-m) x]$ are zero-valued and the integral in Eq. (1.15) is zero. We have therefore shown that:

Equation (1.50) implies that the set of functions given by $\{\sin (n x), \sin (m x)\}$ are mutually orthogonal on $[-\pi, \pi]$.

Similarly, we can show that the functions $1, \cos n x$ and $\sin n x$ are mutually orthogonal on $[-\pi, \pi]$ giving rise to the following:

1. $\int_{-\pi}^{\pi} \cos m x d x=\int_{-\pi}^{\pi} \sin m x d x=0$ if $m$ is integral and $m \neq 0$;
2. $\int_{-\pi}^{\pi} \sin n x \cos m x d x=0$ for any integers $n$ and $m$;
3. $\int_{-\pi}^{\pi} \cos n x \cos m x d x=\int_{-\pi}^{\pi} \sin n x \sin m x d x=0$ if $n \neq m$;
4. $\int_{-\pi}^{\pi} \cos ^{2} m x d x=\int_{-\pi}^{\pi} \sin ^{2} m x d x=\pi$ if $m$ is integral and $m \neq 0$.

These relations are useful in determining the Fourier coefficients.