CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A nonconstant meromorphic function on the plane omits at most two sphere values

Statement

A nonconstant meromorphic function on $C$ omits at most two values of $\hat{C}$ .

Facts & Assumptions

Given: A nonconstant meromorphic function $f : C \to \hat{C}$ .

[L1]

A unique Möbius transformation carries any ordered triple of distinct sphere points to any other (A unique Möbius transformation carries any ordered triple of distinct sphere points to any other).

[L2]

Möbius transformations are biholomorphic sphere self-maps (Every Möbius transformation is a biholomorphism of the Riemann sphere).

[L3]

A nonconstant entire function omits at most one finite value (Little Picard theorem).

Proof

technique · direct

1.1

Assume toward a contradiction that $f$ omits three distinct sphere values. By [L1], choose a Möbius transformation $M$ sending them to $0$ , $1$ , and $\infty$ . Then $g : = M \circ f$ is meromorphic by [L2], omits $0$ , $1$ , and $\infty$ , and therefore is actually entire.

L1L2givenassume-contrachoose

2.1

Fact [L3] makes an entire function omitting $0$ and $1$ constant, so $g$ is constant. Since $M$ is biholomorphic by [L2], $f = M^{- 1} \circ g$ is constant as well, contradicting the hypothesis.

L2L3step 1.1discharge-contradiction

3.1

Therefore a nonconstant meromorphic function on the plane omits at most two sphere values.

step 2.1∎

Depends on

Used by

The exponential function omits 0 and infinity as a meromorphic map on the plane Counterexample
FALSE: a nonconstant meromorphic function on the plane omits at most one sphere value False statement

Dependency tree · two levels

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