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 a_n cosnx + ∑_n^∞₌₁ b_n sinnx   where

a₀  = 1/∏   _c∫ ^{c + 2∏}   f(x) dx

a_n  = 1/∏   _c∫ ^{c + 2∏}   f(x) cosnx dx

b_n  = 1/∏   _c∫ ^{c + 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

a₀  = 1/∏   ₀∫^2∏   f(x) dx

a_n  = 1/∏   ₀∫^2∏   f(x) cosnx dx

b_n  = 1/∏   ₀∫^2∏   f(x) sinnx dx

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

a₀  = 1/∏   -_∏∫^∏   f(x) dx

a_n  = 1/∏   -_∏∫^∏  f(x) cosnx dx

b_n  = 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

-_l∫^l   f(x) dx = 0, if f(x) is an odd function.

-_l∫^l   f(x) dx =2₀∫^l   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 a_n cosnx

a₀  = 2/∏   ₀∫^∏   f(x) dx

a_n  = 2/∏   ₀∫^∏   f(x) cosnx dx

b_n   = 0

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

F(x) =  ∑_n^∞₌₁ b_n sinnx

b_n  = 2/∏   ₀∫^∏   f(x) sinnx dx

a₀  = a_n  = 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 a_n cosn∏x/l + ∑_n^∞₌₁ b_n sinn∏x/l   where

a₀  = 1/l   _c∫ ^{c + 2l}   f(x) dx

a_n  = 1/l   _c∫ ^{c + 2l}   f(x) cosn∏x/l dx

b_n  = 1/l   _c∫ ^{c + 2l}   f(x) cosn∏x/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 a_n cosn∏x/l where

a₀  = 2/l   ₀∫^l   f(x) dx

a_n  = 2/l   ₀∫^l   f(x) cosn∏x/l dx

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

F(x) = ∑_n^∞₌₁ b_n sinn∏x/l   where

b_n  = 2/l   ₀∫^l   f(x) cosn∏x/l dx

Note:

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

F(x) = a₀/2 + ∑_n^∞₌₁ a_n cosnx

a₀  = 2/∏   ₀∫^∏   f(x) dx

a_n  = 2/∏   ₀∫^∏   f(x) cosnx dx

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

F(x) = ∑_n^∞₌₁ b_n sinnx

b_n  = 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^∞_{= -∞}   c_n e^in∏x/l    where

c_{n    =}    1/2l  _c∫ ^{c + 2l}  f(x)   e^-in∏x/l    

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

F(x) = ∑_n^∞_{= -∞}   c_n e^inx    where

c_{n    =}    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 = √_a∫^b [f(x)]²   dx / b-a