THE GEOMETRY OF MATHEMATICAL METHODS

The applet below shows the product (see Section 14.4) 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.

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?

You can check that these periodic functions satisfy the conditions to be interpreted as “vectors” in an abstract vector space (see Section 14.1), and the integral satisfies the conditions to be interpreted as a “dot” or inner product (see Section 14.2). The sines and cosines above are orthogonal! We can therefore use these functions as a natural basis for this vector space of periodic functions. We will discuss this example vector space further in a chapter on Fourier series, Chapter 17.

Periodic functions with sines and cosines as the basis are not the only set of functions that are vector spaces. There are many others, typically associated with a particular differential equation with a particular boundary condition. Each such vector space has a natural inner-product defined on it, similar to (14.6.1) above. The study of these vector spaces as they arise in partial differential equations is called Sturm-Liouville theory.