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On a Riemann surface, the quotient sheaf $\mathcal{P}={\mathcal{M}}_X/{\mathcal{O}}_X$ records the finite negative Laurent series tails of meromorphic germs. At $x$, a local coordinate $t$ identifies its stalk with ${⨁}_{m\ge 1}\mathbb{C}{t}^{-m}$. As a sheaf it is the direct sum of sheaves ${⨁}_x(i_x{)}_{\ast }{\mathcal{P}}_x$. Global sections prescribe locally finite families of principal parts, possibly infinite on a noncompact surface. The connecting homomorphism to $H^1(X,{\mathcal{O}}_X)$ is exactly the obstruction to a global meromorphic function with those parts and no extra poles.

If an autoregressive model has white noise of positive variance, a unique weakly stationary process solving its equation on all integer times exists precisely when its autoregressive polynomial has no root on the unit circle. In that case $1/A(z)$ has an absolutely summable Laurent series on an annulus containing the unit circle; its coefficients define a bilateral linear filter of a stationary time series. Applying that inverse also proves uniqueness among bounded second-moment solutions. A root on the unit circle forces the candidate spectral density of a stationary process to have a nonintegrable singularity. The stronger causality root criterion for an autoregressive model requires every root outside the closed unit disk.

The Weierstrass elliptic function satisfiesWriting $G_{2r}={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-2r}$, the Laurent coefficients of the Weierstrass elliptic function giveThus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the Laurent series of ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.