**Fourier Series**

The greatest inventions of mankind are music and mathematics, and they meet in the theory of Fourier series. A beautiful mathematical theory, which at the same time reveals the interplay between the fundamental frequency and its harmonics, i.e., the mystery of

*timbre*which distinguishes

*music*from

*sound*. The theory originates with

Fourier could not have imagined a form so neat and general, because the first sound theory of integration (Riemann, 1854) was still three decades ahead. This or this single page (not going beyond basic integration) suffices for our proof. Add half a page to obtain the familiar form

valid if the function is (besides mildly regular) also periodic and continuous. In this form we see the fundamental frequency and its infinite number of harmonics displayed. If you want to hear how it

*sounds*, click this image

and play around with the first thirty-two coefficients (blue for cosines, green for sines).

**A mathematician's eulogy**

As any book on the subject will teach you:

*everything about Fourier series is interesting*and there are no easy, let alone*trivial*, results. Pointwise convergence, considered above, is just the first of many, many fascinating aspects. The marvel of the subject was realized early on. In this paper from 1864 Lord Kelvin, the great physicist, describes*Théorie de la Chaleur*as*Fourier's great mathematical poem*

(fourth line of paragraph 4). The quote is frequently attributed to Maxwell, but he doesn't come any closer than "Fourier, in his great work on the conduction of heat..." (

*The scientific letters and papers of James Clerk Maxwell,*ed. by P.M. Harman, Cambridge University Press, 1990-2002, Vol. II, p. 358). But it*is*true that Maxwell read Fourier's book when he was seventeen, and was deeply impressed. I would have preferred the quote to have been Maxwell's, because Maxwell was a talented poet himself. (Speaking of physicists writing poems: Oppenheimer also did.)