10.1. PERIODIC FUNCTIONS

A function f( x) which satisfies the relation f( x + T) = f( x) for all real x and some fixed T is called a periodic function. The smallest positive number T, for which this relation holds, is called the period of f( x).

If T is the period of f( x), then

Also

? f( x) = f( x nT), where n is a positive integer.

Thus, f( x) repeats itself after periods of T.

For example, sin x, cos x, sec x and cosec x are periodic functions with period 2 ? while tan x and cot x are periodic functions with period ?. The functions sin nx and cos nx are periodic with period .

The sum of a number of periodic functions is also periodic. If T ₁ and T ₂ are the periods of f( x) and g( x), then the period of a f( x) + b g( x) is the least common multiple of T ₁ and T ₂.

For example, cos x, cos 2 x, cos 3 x are periodic functions with periods 2 ?, ? and respectively.

? is also periodic with period 2 ?,...