Fourier Series - Chennai Tuition Centre,Home tuition in chennai Chennai Tuition Centre

The General Fourier series for the function f(x) in the interval c < x < c + 2 is given by

F(x) = a0/2 + ∑n=1 an cosnx + ∑n=1 bn sinnx   where

a= 1/   cc + 2∏   f(x) dx

a= 1/   cc + 2∏   f(x) cosnx dx

b= 1/   cc + 2∏   f(x) sinnx dx

 

Note: cosn∏  = (-1)n where n is an integer

           sinn∏  =  0    where n is an integer

 

The Fourier series for the function f(x) in the interval 0 < x < 2 is given by

a= 1/   02∏   f(x) dx

a= 1/   02∏   f(x) cosnx dx

b= 1/   02∏   f(x) sinnx dx

 

The Fourier series for the function f(x) in the interval - < x <   is given by

a= 1/   -∏   f(x) dx

a= 1/   -∏  f(x) cosnx dx

b= 1/   -∏  f(x) sinnx dx

 

Even and odd function:

A function is said to be even if f(-x) = f(x).

A function is said to be odd if f(-x) = - f(x).

 

-∏   f(x) dx = 0, if f(x) is an odd function.

-∏   f(x) dx =20∏   f(x) dx, if f(x) is an even function.

 

Similarly

-ll   f(x) dx = 0, if f(x) is an odd function.

-ll   f(x) dx =20l   f(x) dx, if f(x) is an even function.

 

If a function f(x) is even, its Fourier expansion contain only cosine terms.

F(x) = a0/2 + ∑n=1 an cosnx

a= 2/   0∏   f(x) dx

a= 2/   0∏   f(x) cosnx dx

b = 0

 If a function f(x) is odd, its Fourier expansion contain only sine terms.

F(x) =  ∑n=1 bn sinnx

b= 2/   0∏   f(x) sinnx dx

a0  = a= 0

 

Change of interval:

The  Fourier series for the function f(x) in the interval c < x < c + 2l is given by

F(x) = a0/2 + ∑n=1 an cosnx/l + ∑n=1 bn sinnx/l   where

a= 1/l   cc + 2l   f(x) dx

a= 1/l   cc + 2l   f(x) cosnx/l dx

b= 1/l   cc + 2l   f(x) cosnx/l dx

 

Half range Expansions:

f(x)  defined over the interval 0 < x < l is capable of two distinct half range series.

The half range cosine series in (0,l) is

F(x) = a0/2 + ∑n=1 an cosnx/l where

a= 2/l   0l   f(x) dx

a= 2/l   0l   f(x) cosnx/l dx

 

The half range sine series in (0,l) is

F(x) = ∑n=1 bn sinnx/l   where

b= 2/l   0l   f(x) cosnx/l dx

 

Note:

The half range cosine series in (0,) is

F(x) = a0/2 + ∑n=1 an cosnx

a= 2/   0∏   f(x) dx

a= 2/   0∏   f(x) cosnx dx

 

The half range sine series in (0,) is

F(x) = ∑n=1 bn sinnx

b= 2/   0∏   f(x) sinnx dx

 

Complex or Exponential form of Fourier series:

The complex form of  Fourier series for the function f(x) in the interval c < x < c + 2l is given by

F(x) = ∑n= -∞   cn einx/l    where

cn    =    1/2l  cc + 2l  f(x)   e-inx/l    

 

The complex form of  Fourier series for the function f(x) in the interval - < x <    is given by

F(x) = ∑n= -∞   cn einx    where

cn    =    1/2  -f(x)   e-inx    

 

Root Mean Square value (RMS Value):

R.M.S value of f(x) over the interval (a,b) is defined as

R.M.S = √ab [f(x)]2   dx / b-a

 








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