CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The family e^(nz) converges chordally to infinity on the right half-plane without being holomorphically normal there

Statement refuted

If a holomorphic family converges chordally locally uniformly to $\infty$ , then it is normal in the holomorphic sense of finite-valued locally uniform limits.

Facts & Assumptions

Given: The right half-plane $H = {z \in C : Re z > 0}$ and the sequence $f_{n} (z) = e^{n z}$ .

[L1]

For real $x, y$ , one has $∣ e^{x + i y} ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[L2]

Normal holomorphic families are defined by subsequences with finite-valued holomorphic locally uniform limits (Normal families of holomorphic functions on a plane domain).

Counterexample

technique · direct

1.1

If $K \subseteq H$ is compact, then some $δ > 0$ satisfies $Re z \geq δ$ on $K$ , and [L1] gives $∣ f_{n} (z) ∣ \geq e^{n δ}$ . Therefore $χ (f_{n} (z), \infty) = 2 / \sqrt{1 + ∣ f_{n} (z) ∣^{2}} \leq 2 e^{- n δ}$ on $K$ , so $f_{n} \to \infty$ chordally locally uniformly on $H$ .

L1givenchoosealgebra

2.1

Every subsequence has the same chordal limit $\infty$ , so no subsequence can converge locally uniformly to a finite holomorphic function. By [L2], the family is not normal in the finite-valued holomorphic sense.

L2given∎

Depends on

Used by

FALSE: a chordally locally uniform limit of holomorphic functions can never be identically infinity False statement

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)