Find the Fourier cosine series for f(x) = cos x, 0 < x < Pi
EDIT: I believe we are talking about a half-range extension..
Fourier cosine series:
f(x) = a0/2 + Sum(n = 1 to infinity) (an * cos (nx))
where a0 = 2/L * integral (x = 0, x = L) (f(x) dx)
and an = 2/L * integral (x = 0, x = L) (f(x) * cos (n*Pi*x/L) dx)
Sorry if thats hard to understand...
The Attempt at a Solution
I graphed the function and extended the period to - Pi, i.e. -Pi < x < Pi
I found a0 = 2/Pi * integral (x = 0, x = L) (cos x dx) = 0
an = 2/Pi * integral (x = 0, x = L) (cos x * cos (nx) dx)
and evaluated the integral using IBP...
an = 2/Pi * (ncos(x)*sin (nx) - sin(x)cos(nx))*1/(n2-1)| x = 0, x = Pi
which gives me an = 0!
So the Fourier cosine series would be:
f(x) = 0
I would really appreciate any clarification as getting 0 for the series doesnt sound right..
Thank you in advance