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 
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 
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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