CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The exponential function is meromorphic on C but not meromorphic on the Riemann sphere

Statement refuted

Every meromorphic function on $C$ is meromorphic on the Riemann sphere.

Facts & Assumptions

Given: The exponential function $f (z) = e^{z}$ .

[L1]

Sphere-meromorphic functions are exactly rational functions (Meromorphic functions on the Riemann sphere are exactly the rational functions).

Counterexample

technique · direct

1.1

The function $e^{z}$ is entire on $C$ , so it is meromorphic on $C$ . If it were meromorphic on $\hat{C}$ , then [L1] would make it rational.

L1given

2.1

A rational function with no finite poles is a polynomial, but $e^{z + 2 π i} = e^{z}$ whereas no nonconstant polynomial is $2 π i$ -periodic. Therefore $e^{z}$ is not rational, so it cannot be meromorphic on the sphere.

givenalgebra∎

Depends on

Meromorphic functions on the Riemann sphere are exactly the rational functions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)