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

The required property is odd symmetry, $\boxed{f(-x)=-f(x)}$, up to irrelevant changes on a set of measure zero when discussing Fourier coefficients. An odd function has zero mean, and its products with cosine are odd, so every cosine coefficient vanishes. Its sine coefficients may remain nonzero.

For pointwise representation, the Fourier series takes the average of the one-sided limits at a jump. In particular a pure Fourier sine series has value zero at the origin and at the endpoints; arbitrary assigned values there need not equal the series sum.

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.

Apart from its endpoint convention, the ramp is odd, so its constant and cosine Fourier coefficients vanish. For the sine coefficient, evenness of the integrand and integration by parts giveHenceThis is the Fourier series of a periodic sawtooth. At the periodic jump, the series sums to zero rather than the assigned endpoint value $K/2$; changing a value at one endpoint does not change the coefficients.

Constant translation speed gives $\boxed{\ell (t)=vt}$, since the initial displacement is zero. Substituting into the Fourier series givesThe periodic ramp moves in the positive $x$ direction for $v>0$. At the moving jumps, this series gives the average of the two one-sided values.

Not as ordinary convergent Fourier series, for a nontrivial moving ramp. Formal spatial differentiation would giveand formal time differentiation multiplies that expression by $-v$. The summands have nondecaying amplitude and do not tend to zero, so the ordinary series cannot converge. The periodic extension has jumps, violating the regularity needed for Termwise differentiation of a Fourier series.

Away from the moving jumps, the classical derivatives exist and are $g_x=K/L$ and $g_t=-vK/L$. A distributional derivative also includes the Dirac delta distribution contributions at the jumps, so distributional differentiation is possible, but it is a different interpretation. If $v=0$, the time derivative is trivially zero; if $K=0$, both derivatives are zero.