ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-10-02

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A normal-family proof of Great Picard

Example

Let $f$ be meromorphic on $0 < ∣ z - z_{0} ∣ < r^{*}$ with an isolated essential singularity at $z_{0}$ . The exterior three-value extension lemma rules out three sphere values omitted on any punctured neighbourhood of $z_{0}$ . Consequently every sphere value is attained infinitely often in every punctured neighbourhood, with at most two exceptions. If $f$ is holomorphic there, at most one finite value is exceptional.

Facts & Assumptions

Given: Such a punctured-disc meromorphic function $f$ .

[F1]

If a meromorphic function $G$ on $∣ w ∣ > R$ omits three fixed distinct sphere values on $∣ w ∣ > R_{1}$ for some $R_{1} > R$ , then $G$ extends meromorphically across infinity (Three omitted values force exterior extension).

Verification

technique · invert the puncture and apply the three-value extension lemma

1.1F1givendischarge-contradiction

Suppose three distinct sphere values are omitted on $0 < ∣ z - z_{0} ∣ < ρ$ for some $0 < ρ < r^{*}$ . The function $G (w) : = f (z_{0} + ρ / w)$ is meromorphic on $∣ w ∣ > 1$ and omits the same three values there. By [F1] with $R = 1$ and $R_{1} = 2$ , it extends meromorphically across infinity; inversion then extends $f$ meromorphically across $z_{0}$ , contrary to essentiality.

2.1step 1.1choosecases

If three distinct values each occurred only finitely often in some punctured neighbourhood, take a radius inside all three neighbourhoods and then shrink it below the distance to the finitely many preimages there. If the combined preimage set is empty, any smaller radius works. The three values would all be omitted on the smaller punctured disc, contrary to step 1.1. Hence at most two sphere values are exceptional.

3.1step 2.1algebra∎

If $f$ is holomorphic on the punctured disc, it omits $\infty$ , so step 2.1 leaves at most one exceptional finite value.

Depends on

Three omitted values force exterior extension

Used by

Nothing in the library uses this result yet.

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

Aleksander Simonič, The Ahlfors lemma and Picard's theorems (standard reference, not scraped)