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Periodic Functions

Periodic functions should be familiar to everyone. The keeping of time, the ebb and flow of tides, the patterns and textures of our buildings, decorations, and vestments invoke repetition and periodicity that seem to be inseparable from the elements of human cognition.¹

A function that is periodic in a single variable can be expressed as:

$$\begin{split}f(x + \lambda) &= f(x)\ f(t + \tau) &= f(t) \end{split}$$ (17-1)

The first form is a suggestion of a spatially periodic function with wavelength $\lambda$ and the second form suggests a function that is periodic in time with period $\tau$ . Of course, both forms are identical and express that the function has the same value at an infinite number of points ( $x = n \lambda$ in space or $t = n \tau$ in time where $n$ is an integer.)

Specification of a periodic function, $f(x)$ , within one period $x \in (x_o , x_o + \lambda)$ defines the function everywhere. The most familiar periodic functions are the trigonometric functions:

$$\sin(x) = \sin(x + 2 \pi) \;\; \;\; \cos(x) = \cos(x + 2 \pi)$$ (17-2)

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Making Periodic Functions Periodic functions are often associated with the ``modulus'' operation. Mod$[ x, \lambda ]$ is the remainder of dividing $x$ by $\lambda$ . Its result always lies in the domain $0 \leq$   Mod$[x, \lambda] \leq \lambda)$ . Another way to think of modulus is to find the ``point'' where are periodic function should be evaluated if its primary domain is $x \in (0, \lambda)$ .