Complex Roots — Question 7

Question 7 – Solution

Strategy. Solve before interpreting the data; a sampling grid can repeatedly land on zeros of a nonzero oscillation.

Step 1: Determine all admissible frequencies. The roots are $\pm i\beta$, and the data give $$\boxed{y(t)=\frac{\sin\beta t}{\beta}}.$$ In particular, $y(1)=0$ implies $\sin\beta=0$, so $\beta=m\pi$ for a positive integer $m$. Conversely, each such frequency makes every integer observation zero. The complete candidate set is therefore $$\boxed{\beta=m\pi,\qquad m=1,2,3,\ldots}.$$

Step 2: Retain the slope information correctly. Each candidate has $y'(t)=\cos(m\pi t)$ and hence $y'(0)=1$, so each is nonzero. The zero function would violate that slope. The slope sets the sine coefficient to $1/(m\pi)$, but still allows infinitely many frequencies. A known initial slope does not rescue a schedule that misses all intermediate oscillations.

Step 3: Use the extra observation to select a frequency. For the candidate indexed by $m$, $$y(1/2)=\frac{\sin(m\pi/2)}{m\pi}.$$ For $m=1$ this equals $1/\pi$. For every $m>1$, its absolute value is at most $1/(m\pi)<1/\pi$, making agreement impossible. Thus the new datum selects exactly $\boxed{\beta=\pi}$ and $y=\sin(\pi t)/\pi$.

Step 4: Explain the indistinguishable samples. The integer grid samples an integer number of half-cycles for every candidate, so it always observes a zero. The two plotted curves differ between those times, including in amplitude, yet share all grid values and the initial slope. This is an explicit sampling ambiguity, not a violation of uniqueness: each fixed equation has one solution for its data, while different frequencies define different equations.