Periodic versus non-periodic functions (hw1, ECE301)

Read the instructor's comments here.

Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. More explicitly, a function f is periodic with period P greater than zero if

f(x + P) = f(x) for all values of x in the domain of f. An aperiodic function (non-periodic function) is one that has no such period P (not to be confused with an antiperiodic function (below) for which f(x + P) = −f(x) for some P).

If a function f is periodic with period P, then for all x in the domain of f and all integers n,

f(x + nP) = f(x).

A plot of f(x) = sin(x) and g(x) = cos(x); both functions are periodic with period 2π.A simple example of a periodic function is the function f that gives the "fractional part" of its argument. Its period is 1. In particular,

f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5. The graph of the function f is the sawtooth wave.

The trigonometric functions sine and cosine are common periodic functions, with period 2π. The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.