Example 6.11.8769FF9D2A9B9

Example 6.11.8.769FF9D2A9B9 Let $c\geq 0$ and $\varphi =c\log |z|^2$ on a disc around $0\in \mathbb C$, where $\varphi =0$ if $c=0$. A germ of order $k$ at $0$ is $z^ k$ times a unit, so by polar coordinates it belongs to $\mathcal I(\varphi )_0$ if and only if $\int _0^ rt^{2k-2c+1}\, \mathrm dt<\infty $, that is, $k+1>c$. Hence $\mathcal I(\varphi )=(z^{\lfloor c\rfloor })$. When $c$ is a positive integer, the germ $z^{c-1}$ just fails, because the radial integral diverges logarithmically. This strict threshold shows why theorem 6.11.4 concerns increasing sequences: on the unit disc, the weights $\varphi _ j=(1-1/j)\log |z|^2$, $j\geq 2$, decrease to $\log |z|^2$, yet $\mathcal I(\varphi _ j)=\mathcal O$ for every $j$, whereas $\mathcal I(\log |z|^2)=(z)$.