Fourier Series : Notes with Examples - IIT JAM PDF Download

Periodic Functions and Trignometric Series

Periodic Functions

A function f (x) is called periodic if for all real x in the domain of f (x) there is some positive number p such that
f (x + p ) = f (x) for all x .
The number p is called period of f (x) . The graph of such function is obtained by periodic repetition of its graph in any interval of length p .
NOTE:

(i) Familiar periodic functions are sine and cosine functions.

(ii) The function f = constant is also a periodic function.

(iii) The functions that are not periodic are x, x², x³, e^x, cosh x, ln x etc.

(iv) ∵ f (x+2 p) = f [(x + p)+p ] = f (x + p)=f (x)
Thus for any integer n , f (x + np) = f (x). Hence 2p ,3p,...... are also period of f (x) .
(v) If f (x) and g (x) have period p , then the function
h (x) = af (x) + bg (x)    (a, b constants)
has also period p .

Fundamental Period

If a periodic function f (x) has a smallest period p (> 0) , this is often called the fundamental period of f (x).

Example:
(i) For sin x and cos x the fundamental period is 2π .

(ii) For sin 2x and cos 2x the fundamental period is π.

(iii) For tan x and cot x the fundamental period is π.

(iv) A functions without fundamental period is f = constant.

Trignometric Series

These functions have period 2π. Figure below shows the first few of them.