Complex Fourier Coefficients

E_BRAH

Can anyone help me with this exam question?


Calculate the complex Fourier coefficients[latex] $C_n$ [/latex] for

[latex]$$f(x)= |x|- \frac{\pi}{2} , |x|\leq \pi $$[/latex], and [latex]$2\pi$[/latex] periodically continued to all [latex]$x$[/latex].

I have tried the following;

[latex]$C_n = \frac{1}{2\pi} {\displaystyle \int_{-\pi}^{\pi} f(x)e^{-inx}dx}$

$ = \frac{1}{2\pi} \Bigg( {\displaystyle \int_{-\pi}^{0} (-x-\frac{\pi}{2})e^{-inx}dx} + {\displaystyle \int_{0}^{\pi} (x-\frac{\pi}{2})e^{-inx}dx} \Bigg)$ [/latex]

When I work down through these integrals, I get [latex] $ \displaystyle\frac{(-1)^n}{n^2\pi}$[/latex], whereas I think the answer should be [latex]$\displaystyle \frac{(-1)^n-1}{ n^2\pi}$[/latex]

Is there something wrong with my approach or is it right and my computation is wrong?

Thanks