Let $(\rho_0,\rho_1)$ be a smooth <partition of unity> subordinate to $(U_0,U_1)$. On $U_0$, extend $\rho_1\varphi$ by zero away from the overlap, and on $U_1$ extend $\rho_0\varphi$ similarly. Define
$$
\psi_0=\exp(\rho_1\varphi)\quad\hbox{on }U_0,
\qquad
\psi_1=\exp(-\rho_0\varphi)\quad\hbox{on }U_1.
$$
On the overlap,
$$
\frac{\psi_1}{\psi_0}=e^{-(\rho_0+\rho_1)\varphi}=e^{-\varphi}=h^{-1},
$$
and therefore
$$
\frac{\psi_1}{\psi_0}f=h^{-1}f=z^k.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / h / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-115/2/h/solution