Fourier coefficient
= Fourier coefficient
{c}
{title2=$\widehat f_n$}
For an integrable <function> of <period> $L>0$, its complex <Fourier coefficients> measure its <inner product> with each <complex exponential> mode:
$$
\widehat f_n=\frac1L\int_{x_0}^{x_0+L}f(x)e^{-2\pi inx/L}\,dx.
$$
For a real <function> of period $2\pi$, the real coefficient convention of <Fourier series> gives $\widehat f_0=a_0/2$ and $\widehat f_n=(a_n-ib_n)/2$ for $n\geq1$, with $\widehat f_{-n}=\overline{\widehat f_n}$. For $f(x)=1/2+\cos x+\tfrac12\sin(2x)$, the only nonzero complex <Fourier coefficients> are $\widehat f_0=1/2$, $\widehat f_{\pm1}=1/2$, $\widehat f_2=-i/4$, and $\widehat f_{-2}=i/4$. This explains why the real and complex lists look different while representing the same <function>.