Question 366527: find fourier series of cos^3 of x, and sin^3 of x ..
any one could help ..pleez,,
thanks Answer by Jk22(389) (Show Source):
You can put this solution on YOUR website! Given a function f(x) , we shall write the Fourier series of this.
There are series in complex exponential, sine or cosine. Let f(x) a function which is 2Pi periodic
is the series in cosine,
with
since cos(mx) and cos(nx) are orthogonal functions.
Consider an even function like cos(x)^3, only cosine (even) terms will appear.
However without doing the integration, we can write cos(x) with complex numbers : cos(x) = (exp(ix)+exp(-ix))/2
then cos(x)^3 = 1/8*(exp(i3x)+3exp(ix)+3exp(-ix)+exp(-i3x))
= cos(3x)/4 + 3/4*cos(x)
We can now show that this corresponds to the definition above :
hence
The same can be shown for :
the similar calculation can be done with sin(x) = (exp(ix) - exp(-ix))/(2i)
