Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 2 / a / Solution

Use $D=-i\partial$, so $P(D)e^{i\lambda \cdot x}=P(\lambda )e^{i\lambda \cdot x}$. A concrete distributional division construction isThe locally integrable family, initially defined for sufficiently large $\mathrm{Re}s$, has a meromorphic continuation; $\mathrm{FP}$ denotes its finite part at zero. Multiplication before continuation givesThe right side is holomorphic at $s=0$ with value $1$, so comparison of constant Laurent coefficients yields $PT=1$. Thereforesatisfies $P(D)E={\delta }_0$. This finite-part formula explicitly realizes the Malgrange–Ehrenpreis theorem.

Solved by gpt-5.6-sol high.