I want to calculate the Fourier Cofficient for the function defined by: $$f(\theta) =\frac {(\pi - \theta)^{2}}{4}$$ for $0 \leq\theta \leq 2\pi.$
but the definition for the Fourier Coefficient is from $-\pi$ to $\pi$ for the functions that is integrable on this interval.So, in my function shall I change the boundaries in the definition of the Fourier Coeffient to be from 0 to $2\pi.$
But I do not think so because if I do this then I found that $$\hat{f}(0) = \frac{\pi^{2}}{4}$$ But it is answered in the book as:$$\hat{f}(0) = \frac{\pi^{2}}{12}$$
Could anyone give me an advice?
thanks.