Integrals of Sines and Cosines

Last week in this blog, I discussed the Fourier series. This week, I want to highlight some remarkable mathematical formulas that make the Fourier series work: integrals of sines and cosines. The products of sine and cosines obey these relationships:

where n and m are integers. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I dedicate Appendix E to studying these integrals. They allow some very complicated expressions involving infinite sums to reduce to elegantly simple equations for the Fourier coefficients. Whenever I’m teaching Fourier series, I go through the derivation up to the point where these integrals are needed, and then say “and now the magic happens!”

The collection of sines and cosines (sinmx, cosnx) are an example of an orthogonal set of functions. How do you prove orthogonality? One can derive it using the trigonometric product-to-sum formulas.

I prefer to show that these integrals are zero for some special cases, and then generalize. Russ and I do just that in Figure E2. When we plot (a) sinx sin2x and (b) sinx cosx over the range 0 to 2π, it becomes clear that these integrals are zero. We write “each integrand has equal positive and negative contributions to the total integral,” which is obvious by merely inspecting Fig. E2. Is this a special case? No. To see a few more examples, I suggest plotting the following functions between 0 and 2π:

In each case, you will see the positive and negative regions cancel pairwise. It really is amazing. But don’t take my word for it, as you’ll miss out on all the fun. Try it.

Nearly as amazing is what happens when you analyze the case for m = n by integrating cosnx cosnx = cos²nx or sinnx sinnx=sin²nx. Now the integrand is a square, so it always must be positive. These integrals don’t vanish (although the “mixed” integral cosnx sinnx does go to zero). How do I remember the value of this integral? Just recall that the average value of either cos²nx or sin²nx is ½. As long as you integrate over an integral number of periods, the result is just π.

When examining non-periodic functions, one integrates over all x, rather than from merely zero to 2π. In this case, Russ and I show in Sec. 11.10 that you get delta function relationships such as

I won’t ask you to plot the integrand over x, because since x goes from negative infinity to infinity it might take you a long time.

The integrals of products of sines and cosines is one example of how Russ and I use appendicies to examine important mathematical results that might distract the reader from the main topic (in this case, Fourier series and its application to imaging), but are nevertheless important.