= Translation of a Fourier series
For a period-$L$ <Fourier series> written as $a_0/2+\sum_{n\geq1}(a_n\cos k_nx+b_n\sin k_nx)$, where $k_n=2\pi n/L$, translating the <function> by $\ell$ to $g(x)=f(x-\ell)$ rotates each pair of <Fourier coefficients>:
$$
a_n^{(g)}=a_n\cos(k_n\ell)-b_n\sin(k_n\ell),\qquad
b_n^{(g)}=a_n\sin(k_n\ell)+b_n\cos(k_n\ell).
$$
The constant coefficient is unchanged. These formulas follow by the sine and cosine subtraction identities. In the <complex Fourier series>, the same statement is $\widehat g_n=e^{-ik_n\ell}\widehat f_n$. An integer-period translation changes no <Fourier coefficient>.
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