Inner Products of Functions

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Section 9.5 Inner Products of Functions

The applet below shows the product (see Section 9.4) of two trigonometric functions whose periods are related by a factor of \(2\text{.}\) There is about as much area above the graph of this product as below, so it is plausible that the integral of this product should be \(0\text{,}\) which turns out to be correct. This result holds for the product of any two sine or cosine functions whose periods are related by a rational number other than \(1\text{,}\) so long as the integral is taken over (a multiple of) a full period of both functions. ¹

You can verify this remarkable fact by inserting any two distinct integers into the functions in the applet, and/or changing one or both functions from cosine to sine. What happens if you use the same integer twice?

If we interpret periodic functions as “vectors” in an abstract vector space (see Section 9.1), and the integral as the “dot product”, then these functions are orthogonal! We can therefore use these functions as a natural basis for this vector space of periodic functions.

Figure 9.5.1. The product of two trigonometric functions.

More formally, there is an inner product (see Section 9.2) on (suitably smooth, periodic) functions given by ²

\begin{equation} \langle f|g\rangle = \int_0^L f(x)^*g(x)\,dx\tag{9.5.1} \end{equation}

under which the functions \(\left\{1,\cos\left(\frac{2m\pi x}{L}\right), \sin\left(\frac{2m\pi x}{L}\right)\right\}\) are orthogonal. However, they are not normalized, as you can verify using the applet above.

In the applet shown, with the integral being taken from \(0\) to \(1\text{,}\) this condition is equivalent to requiring the coefficients of \(\pi x\) to be distinct even integers. (The integers can be the same if one function involves sine and the other cosine.)

The “bra-ket” notation used here is often used in quantum mechanics and is further discussed in Section 9.2. In the context of real functions, such as those considered here, the complex conjugation doesn't do anything, but is included anyway to better match the notation used in more general contexts.