Periodic function

These are the function, whose value repeats after a fixed constant interval called period, and which makes a class of a widely used function.

        A function f of x, such that:

        f(T + x) = f(x) ∀ x ε domain of f.

        The least positive real value of T for, which above relation is true, is called the fundamental period or just the period of the function.

        e.g. for f(x) = sin x ∀ x ε R.

We know that sin (2∏ + x) = sin x, ∀ x ε R

so f(x) = sin x is a periodic function with a period of 2∏ radians.

(i)     If f(x) is periodic with period p, then a f(x) + b, where a, b ε R (a≠0) is also a periodic function with period p.

(ii)    If f(x) is periodic with period, then f(ax + b), where a ε R -{0} and b ε R, is also periodic with period p/|a|.

(iii)    let us suppose that f(x) is periodic with period p and g(x) is periodic with period q. Let r be the L.C.M. of p and q, if it exists.

(a)    If f(x) and g(x) cannot be interchanged by adding a least positive number k, then r is the period of f(x) + g(x).

(b)    If f(x) and g(x) can be interchanged by adding a least positive number k and if k < r, then k is the period of f(x) + g(x). Otherwise r is the period.