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M2

Unit-5

Fourier series

 

Q1: Find the fourier series of the function f(x) = x      where 0 < x < 2 π

 

A1:
We know that, from fourier series,

 

First we will find ,

Now,

 

 

And ,

 

 

Put these value in Fourier series, we get

 

Q2: Find the Fourier series for f(x) = x / 2 over the interval 0 < x < 2π

And has period 2π

 

A2:

First we will find

=

=

= π

   =   π

Similarly,

 

 

 

Which gives,    =   0

Now,


 

 

 

We get,

We know that, the Fourier series

 

Put these values in Fourier series, we get

 

Q3: Find the Fourier series for f(x) = in the interval .

A3:

Suppose

Then-

And

 

So that-

And then-

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

 

Q4: Find the fourier series of the function f(x) = x      where 0 < x < 2 π

 

A4:

We know that, from fourier series,

f(x) =

 

first we will find ,

Now,

And ,

Put these value in fourier series, we get

 

Q5: Find the fourier series for f(x) = x / 2 over the interval 0 < x < 2π

And has period 2π

 

A5:

First we will find

  = 

                                                   =

                                                   =

                                                  = π

   =   π

Similarly,

 

Which gives,    =   0

Now,

 


 

 

We get,

We know that, the fourier series

 

Put these values in fourier series, we get

 

Q6: Find the Fourier series for f(x) = in the interval .

A6:

Suppose

Then-

And

 

So that-

And then-

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

Q7: Find the fourier expression of f(x) = x³ for –π< x <π.

A7:

Here, we can see that f(x) Is an odd function

So that,

and

We will use here ,

We get the value of f(x),

 

Q8: Find a Fourier series for

; 

A8:

Here

; 

Since f(x) is even function hence

It’s Fourier series is

     … (1)

Where

Hence equation (1) becomes,

 

Q9: Find the Fourier sine series for the function-

Where ‘a’ is a constant.

A9:
here we know-

We know that-

And


Q10: Using complex form,find the Fourier series of the function

f(x) = sinx =

A10:

We calculate the coefficients

 

 

 

 

 

  =

 

=

Hence the Fourier series of the function in complex form is

 

We can transform the series and write it in the real form by renaming as

 n=2k-1,n=

 

 

 

 

=