I'm asked to find the Fourier series expansion, valid on $]-\pi,\pi[$ for the function:
$f(x)= \left\{ \begin{array}{ll} 0 & -\pi < x\leq 0 \\ cos(x) & 0< x <\pi\\ \end{array} \right. $
And also the sum of the series for $x=p \cdot \pi$ for $p \in \mathbb{Z}$.
Now; I understand that the function has a 'jump discontinuity' at x=0, where $cos(0)=1 \neq 0$. And the average value here, which the Series converges to is $\frac{f(x^+)+f(x^-)}{2}=\frac{cos(0)+0}{2}=\frac{1}{2}$.
But I am fairly inexperienced with such series, and I don't feel that my textbook elucidates it very well. For finding the coefficients... do I add $\frac{1}{2}$? Or is it simply that $a_0=\frac{1}{2}$ and the remaining $a_n$'s are zero? (Since $\int_{0}^{\pi}cos(x)\cdot cos(nx) = 0$)
And with regards to $b_n$ does it follow the normal formula? I.e. $b_n=\int_{0}^{\pi}cos(x)\cdot sin(nx)$?
Any help much appreciated. Thanks for reading.