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For $X={\mathbb{P}}^1\times \mathbb{C}$, the two standard affine charts and their intersection form an acyclic cover for the sheaves of holomorphic forms. Their Čech cohomology gives $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$ and $H_{\bar{\partial }}^{2,1}=\mathcal{O}(\mathbb{C})$, with all other groups for $p\in \{0,2\}$ zero. For the top-form sheaf, the transition $d(1/z)=-z^{-2}dz$ removes every Laurent series power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary entire function on the affine factor.

An even function satisfies $f(-x)=f(x)$; its Laurent series or taylor series contains only even powers.

For $x\in \mathbb{R}$ and $R>0$, the filled half-disc $K=\{z\in \mathbb{H}:|z-x|\le R\}$ is a compact H-hull. Its mapping-out function of a compact H-hull is $g_K(z)=z+R^2/(z-x)$. On the semicircle the image is real, and the inverse branch asymptotic to the identity maps the complex upper half-plane onto the exterior half-disc. Its Laurent series has coefficient $R^2/z$, so $\boxed{\mathrm{hcap}(K)=R^2}$. Using an open half-disc instead requires taking its relative closure first.

For $X={\mathbb{C}}^2\setminus \{0\}$, the cover $\mathbb{C}\times {\mathbb{C}}^{\ast }$ and ${\mathbb{C}}^{\ast }\times \mathbb{C}$ is acyclic for ${\mathcal{O}}_X$. The first Čech cohomology quotient is represented uniquely by normally convergent doubly negative Laurent series. Under $w_i=1/z_i$ it is $w_1{w}_2\mathcal{O}({\mathbb{C}}_w^2)$, the entire functions divisible by both coordinates. This is an infinite-dimensional analytic vector space; restricting representatives to finite Laurent polynomials loses classes. The class of $1/(z_1{z}_2)$ is nonzero, so $X$ is not a Stein manifold.

An odd function satisfies $f(-x)=-f(x)$; its Laurent series or taylor series contains only odd powers.