Periodic Function

A function f(x) is periodic if f(x) = f(x + p) for all x and some fixed p. p is called a period of f. The smallest positive p that works is called the period of f. (A constant function is periodic with any value of p as a period, so there's no such thing as "the period" of such a function. Any continuous non-constant periodic function has a unique smallest positive period.)

For example, the cosine function is periodic with period 2pi.

In some sense, that single example tells the whole story. For any sufficiently nice periodic function (for instance, one that has only finitely many discontinuities within any finite interval) can be written as an infinite sum of sines and cosines. More precisely, if f is periodic with period p then f(x) = sum for n from 0 to infinity of a(n)*cos(nx) + b(n)*sin(nx). (This can be written more elegantly in terms of complex exponentials.) This stuff was first worked on by Jean-Baptiste Joseph Fourier, so it's called FourierAnalysis (see ComplexFourierSeries).

CategoryMath