Question

I know if f(x) is even the fourier series expansion will consists of consnx, for like f(x)=x^2sinx, or f(x)=2/(3+cosx). but if the f(x) is neither even or odd, would fourier expansion have both cosnx and sinnx? This is PDE.

Answer 1

The Fourier series of a function on the interval _{(here is periodic of period )} is as follows:

   ………………(1)

Where are constants need to be determined.

Now integrate (1) both sides with respect to from to we have,,

this implies   

Now to find ; Multiply both side of equation (1) and integrate from to we have,

this implies

  

Similarly to find ; we multiply in (1) and integrate with respect to from to , we get

  

NOTE: are Fourier coefficients and is called Fourier series of , . This the Most general case, whereas the even and odd function expansion can be derive from here as follows.