TheoremStatement: Literature-sourcedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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Great Picard theorem

Statement

Assume the Axiom of Choice. Let $f$ be holomorphic on a punctured disc $0 < ∣ z - a ∣ < R$ and suppose $a$ is an essential singularity of $f$ . With at most one finite exception, every value in $C$ is assumed infinitely often in every punctured neighborhood of $a$ .

Facts & Assumptions

Given: The Axiom of Choice and a holomorphic function on $0 < ∣ z - a ∣ < R$ with an essential singularity at $a$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it is used in the omitted-values lemma to select a convergent subsequence from a normal family.

[L1]

Under the Axiom of Choice, if a punctured-disc holomorphic function omits two distinct finite values, then the singularity is removable or a pole (Two omitted finite values rule out an essential singularity).

Proof

technique · direct

1.1A1L1givenassume-contradischarge-contradiction

Suppose two distinct finite values $w_{1}, w_{2}$ each failed to occur infinitely often in some punctured neighborhood of $a$ . After passing to the smaller of those neighborhoods, each equation $f (z) = w_{j}$ would have only finitely many solutions there. Shrink once more past all those finitely many points. The resulting punctured disc omits both $w_{1}$ and $w_{2}$ , so [L1], under [A1], would make the singularity removable or a pole, contradicting the hypothesis that it is essential.

2.1step 1.1algebra

Step 1.1 shows that at most one finite value can fail the asserted infinitely-often property. Every other finite value is therefore assumed infinitely often in every punctured neighborhood of $a$ .

3.1step 2.1∎

This is exactly the Great Picard conclusion for finite values.

Depends on

Used by

Dependency tree · two levels

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Sources

Aleksander Simonic, The Ahlfors lemma and Picard's theorems, §6.4 (standard reference, not scraped)