 Translation of a Fourier series

For a period-$L$ Fourier series written as $a_0/2+{\sum }_{n\ge 1}(a_n\cos k_{n}x+b_n\sin k_{n}x)$, where $k_n=2\pi n/L$, translating the function by $\ell$ to $g(x)=f(x-\ell )$ rotates each pair of Fourier coefficients:The constant coefficient is unchanged. These formulas follow by the sine and cosine subtraction identities. In the complex Fourier series, the same statement is ${\hat{g}}_n=e^{-i{k}_n\ell }{\hat{f}}_n$. An integer-period translation changes no Fourier coefficient.