Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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FALSE: an entire function bounded on the real axis is constant

Statement

False claim: If an entire function is bounded on the real axis, then it is constant.

Facts & Assumptions

Given: The complex sine function.

[L1]

The complex sine and cosine power series have infinite radius of convergence (The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series).

[L2]

Complex sine restricts to the published real sine function on the real axis (The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over C).

[L3]

For every real x, ∣sin⁡x∣≤1 (Parity and the Pythagorean identity for sine and cosine).

[L4]

Complex sine and cosine are entire and satisfy sin⁡′=cos⁡ and cos⁡′=−sin⁡ (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives).

Refutation

technique · direct
1.1

By [L1] and [L4], complex sine is entire; by [L2] and [L3], its restriction to the real axis satisfies ∣sin⁡x∣≤1 for every real x.

L1L2L3L4
1.2

By [L4], the derivative of sine is cosine, and the series in [L1] gives cos⁡0=1, so sine has a nonzero derivative and is not constant.

L1L4algebra
2.1

Steps 1.1 and 1.2 give an entire function bounded on the real axis but not constant, refuting the claim; this does not contradict Liouville's theorem: every bounded entire function is constant, whose hypothesis is boundedness on the whole complex plane.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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