 Fourier coefficient

For an integrable function of period $L>0$, its complex Fourier coefficients measure its inner product with each complex exponential mode:For a real function of period $2\pi$, the real coefficient convention of Fourier series gives ${\hat{f}}_0=a_0/2$ and ${\hat{f}}_n=(a_n-i{b}_n)/2$ for $n\ge 1$, with ${\hat{f}}_{-n}=\overline{{\hat{f}}_n}$. For $f(x)=1/2+\cos x+\frac{1}{2}\sin (2x)$, the only nonzero complex Fourier coefficients are ${\hat{f}}_0=1/2$, ${\hat{f}}_{\pm 1}=1/2$, ${\hat{f}}_2=-i/4$, and ${\hat{f}}_{-2}=i/4$. This explains why the real and complex lists look different while representing the same function.