SOLUTION: find fourier series of cos^3 of x, and sin^3 of x .. any one could help ..pleez,, thanks

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)

sin(x)^3 = (exp(3ix) - 3exp(ix) + 3exp(-ix) - exp(-i3x))/(2i)^3

= (- sin(3x) + 3sin(x))/4
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