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# What is the periodic function?

## Overview(periodic function)

A function f(x) is said to be periodic function if f(x + T) = f(x) for all real x and some positive number T, here T is the period of f(x).

Suppose if we take sin x, then it repeats its value after the period of 2pi

such that, we write this as

We can say that sin x is a periodic function with the period of .

sin x, cos x, sec x, cosec x are the periodic functions with period , where tan x and cot x are the periodic functions with period

Note

1.     Sin x is also known as sinusoidal periodic function.

2.     The period of a sum of a number of periodic functions is the least common multiple of the periods.

3.     A constant function is periodic for any positive T.

4.     Suppose T is the period of f(x) then nT is also periodic of f for any integer n.

## Fourier series

Trigonometric series-

This is a functional series of the form,

The constants a0, an and bn are the coefficients.

Fourier series of a periodic function f(x) with period is the trigonometric series with Fourier coefficients  a0, an and bn

Any periodic function can be expanded in the form of Fourier series.

How to determine

We know that, the Fourier series,

To find

Intergrate equation (1) on both sides, from 0 to 2π

That gives

To find

Multiply each side of eq. (1) by cos nx and integrate from 0 to 2π

We get,

Similarly we can find by, multiplying eq. (1) by sin nx and integrating from 0 to  2π

## Solved example

Example: Find the fourier series of the function f(x) = x  where 0 < x < 2 π

Sol. We know that, from Fourier series,

First we will find a_0

Now a_n

And

Now put the value in Fourier series, we get

Example: Find the Fourier series for f(x) = in the interval

Sol.

Suppose

Then

And

So that

And then

Now put these value in equations (1), we get-

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