Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 18 / b

Use the periodic extension of $f$ when the translated argument falls outside the original interval. With $k_n=2\pi n/L$, the sine subtraction identity givesThis translation of a Fourier series generally introduces cosine terms: a translated odd function need not be odd about the original origin. Equality is understood wherever the original Fourier series represents its periodic extension, with midpoint values at jumps.

If $\ell =mL$ with integer $m>0$, then $k_n\ell =2\pi nm$. All sine phase factors vanish and all cosine phase factors equal one. Thereforefor the periodic extension, and every Fourier coefficient is unchanged.

Write the translated Fourier series in the convention $a_0^{(g)}/2+{\sum }_{n\ge 1}(a_n^{(g)}\cos k_{n}x+b_n^{(g)}\sin k_{n}x)$. Comparing with (i) givesThe minus sign in the cosine coefficient records translation in the positive $x$ direction.