The holomorphic function $J(w)=-(w+w^{-1})$ maps $D_{+}=\{w:|w|<1,\mathrm{Im}w>0\}$ conformally and bijectively onto the upper half-plane. Its imaginary part is $\mathrm{Im}w(|w{|}^{-2}-1)>0$, and its derivative $w^{-2}-1$ never vanishes there.

Equality of two values implies $(w_1-w_2)(1-1/(w_1{w}_2))=0$; since $|w_1{w}_2|<1$, this proves injectivity. For any target $\zeta$ in the upper half-plane, the quadratic equation $w^2+\zeta w+1=0$ has two roots with product one and neither on the unit circle. Exactly one therefore has modulus below one; the imaginary-part formula forces that root to lie in $D_{+}$, proving surjectivity.

In particular, $z\mapsto e^{iz}$ maps $\{0<\mathrm{Re}z<\pi ,\mathrm{Im}z>0\}$ bijectively to $D_{+}$, and composition gives the conformal map $z\mapsto -2\cos z$ onto the upper half-plane. The two sides of the half-strip map to the outer real rays, while its bottom edge maps to the segment $[-2,2]$.

Represent a generated sequence by shared memoized lazy list cells and let each iterator hold its own current cell reference. Copy that reference into a new iterator object. Advancing one cursor changes no other cursor, while every cell's future tail is generated once. Thus even a privately stateful generator can supply identical future values to independently consumed snapshots, provided its state is owned by this shared sequence and not mutated externally. Retained older cursors can keep cached prefixes alive. ^[ref]

The +2 oxidation state of iron; the aqueous cation is commonly represented by $\mathrm{F}{\mathrm{e}}^{2+}$.

The +3 oxidation state of iron; the aqueous cation is commonly represented by $\mathrm{F}{\mathrm{e}}^{3+}$.

A family of iron hydroxide materials. Their precipitation can obscure colour tests unless the analytical sequence controls the pH.

A chemical compound containing iron; the oxidation state and associated ligands distinguish medicinal preparations.

The sulfide represented by $\mathrm{FeS}$, used historically to generate hydrogen sulfide with dilute nonoxidizing chemical acids.

The ferrous hydroxide material represented by $\mathrm{Fe}(\mathrm{OH}{)}_2$, which oxidizes on exposure to air.

The hydrated ferric precipitate conventionally represented as $\mathrm{Fe}(\mathrm{OH}{)}_3$ in qualitative analysis; real precipitates need not consist of one uniquely defined pure hydroxide phase.

Iron wire is iron-containing material drawn into a slender form with a large length-to-cross-section ratio. Its behavior depends on purity, alloy composition, surface oxide, and mechanical history.

The geometry gives a high exposed area relative to bulk pieces and determines electrical resistance and heat transfer. An experiment using iron wire therefore needs both material identity and dimensions; the word iron alone does not specify the whole experimental object.

The sphincter pupillae muscle is the circular smooth muscle of the iris that constricts the pupil. Its activation produces miosis. It differs from the radial dilator both in fiber arrangement and in its autonomic control pathway.

The opening in the iris through which light enters the eye. Its aperture affects admitted pencils of rays; the cornea and crystalline lens provide focusing power.

The iris dilator muscle has a roughly radial arrangement that enlarges the pupil when activated. Its action opposes that of the sphincter pupillae muscle. Autonomic control links the size of the optical opening to physiological state.

Bowel is a common anatomical term for the intestine. Context should specify small or large bowel when their different functions matter. The broad word does not identify one single segment or uniform mucosal structure.

The colon is the main elongated portion of the large intestine between the cecal region and rectum. Its segments support water and electrolyte handling, microbial processing and movement of contents. Anatomical position and transit behavior vary along its length.

Intestinal obstruction is impairment of onward passage through the intestine. Mechanical blockage and failure of coordinated propulsion are different routes to impaired transit. Distension and upstream accumulation describe consequences, not a unique cause.

A Peyer's patch is an organized lymphoid aggregation in the intestinal wall, particularly prominent in the ileum. It supports sampling and immune responses associated with gut contents. Its local position distinguishes it from a draining mesenteric lymph node.